Logarithm

Overview

**logarithm**of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if

*x*=

*b*

^{y}, then

*y*is the logarithm of

*x*to base

*b*, and is written log

_{b}(

*x*), so log

_{10}(1000) = 3.

Logarithms were introduced by John Napier

John Napier

John Napier of Merchiston – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer...

in the early 17th century as a means to simplify calculations.

Unanswered Questions

Encyclopedia

The

Logarithms were introduced by John Napier

in the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily and rapidly, using slide rule

s and logarithm table

s. These devices rely on the fact—important in its own right—that the logarithm of a product

is the sum

of the logarithms of the factors:

The present-day notion of logarithms comes from Leonhard Euler

, who connected them to the exponential function

in the 18th century.

The logarithm to base

and has many applications in science and engineering. The natural logarithm

has the constant

(≈ 2.718) as its base; its use is widespread in pure mathematics

, especially calculus

. The binary logarithm

uses base

.

Logarithmic scale

s reduce wide-ranging quantities to smaller scopes. For example, the decibel

is a logarithmic unit quantifying sound pressure

and voltage ratios. In chemistry, pH

is a logarithmic measure for the acid

ity of an aqueous solution

. Logarithms are commonplace in scientific formula

s, and in measurements of the complexity of algorithms

and of geometric objects called fractal

s. They describe musical intervals

, appear in formulas counting prime number

s, inform some models in psychophysics

, and can aid in forensic accounting

.

In the same way as the logarithm reverses exponentiation

, the complex logarithm

is the inverse function

of the exponential function applied to complex numbers. The discrete logarithm

is another variant; it has applications in public-key cryptography

.

, that is raising a number to a power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2:

It follows that the logarithm of 8 with respect to base 2 is 3.

, is done by multiplying

The

. For example,

of

The logarithm is denoted "log

since

A third example: log

Typical scientific calculators calculate the logarithms to bases 10 and

. Logarithms with respect to any base

Given a number

(the irrational

mathematical constant ≈ 2.71828), and

, the logarithm to base

number system:

Thus, log

strictly bigger than log

, where the binary system

is ubiquitous.

The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write log(

(ISO 31-11

).

worked with the concept of

published

in 1544 which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.

In the 16th and early 17th centuries an algorithm called prosthaphaeresis

was used to approximate multiplication and division. This used the trigonometrical identity

or similar or convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.

The Babylonians

sometime in 2000–1600 BC invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of squares. However it could not be used for division without an additional table of reciprocals. This method was used to simplify the accurate multiplication of large numbers till superseded by the use of computers.

in 1614, in a book entitled

independently invented logarithms but published six years after Napier.

By repeated subtractions Napier calculated for

Napier first called

where the very close approximation corresponds to the observation that

The invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri

(Italy), Edmund Wingate

(France), Xue Fengzuo (China), and

Johannes Kepler

's

In 1647 Grégoire de Saint-Vincent

related logarithms to the quadrature of the hyperbola, by pointing out that the area

The natural logarithm was first described by Nicholas Mercator

in his work

defined the exponential function and the natural logarithm by

Euler also showed that the two functions are inverse to one another.

. They were critical to advances in surveying

, celestial navigation

, and other domains. Pierre-Simon Laplace

called logarithms

A key tool that enabled the practical use of logarithms before calculators and computers was the

in 1617, immediately after Napier's invention. Subsequently, tables with increasing scope and precision were written. These tables listed the values of log

and

For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis

, which relies on trigonometric identities. Calculations of powers and roots

are reduced to multiplications or divisions and look-ups by

and

Many logarithm tables give logarithms by separately providing the characteristic and mantissaThe common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...

of

of log

Another critical application was the slide rule

, a pair of logarithmically divided scales used for calculation, as illustrated here:

The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred

enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of less precision, much faster computation than techniques based on tables.

)

has a solution

from elementary calculus

. This theorem states that a continuous function

which produces two values

This property can be shown to hold for the function

(for ), or strictly decreasing (for ).

The unique solution

In prose, taking the power of

says that first taking the logarithm and then exponentiating gives back

Inverse functions are closely related to the original functions. Their graphs

correspond to each other upon exchanging the

, so is log

of

, the chain rule

implies that the derivative of log

That is, the slope

of the tangent

touching the graph of the logarithm at the point equals . In particular, the derivative of ln(

of 1/

The quotient at the right hand side is called the logarithmic derivative

of

Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.

of 1/

In other words, ln(

and the fact that derivative of ln(

The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (

The power formula ln(

The second equality uses a change of variables (integration by substitution

),

The sum over the reciprocals of natural numbers,

is called the harmonic series

. It is closely tied to the natural logarithm: as

, the difference,

converges

(i.e., gets arbitrarily close) to a number known as the Euler–Mascheroni constant

. This relation aids in analyzing the performance of algorithms such as quicksort.

based on repeated squarings of

Newton's method

, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently. Using look-up tables, CORDIC

-like methods can be used to compute logarithms if the only available operations are addition and bit shifts

.

From a theoretical point of view, the Gelfond–Schneider theorem

asserts that logarithms usually take "difficult" values. The formal statement relies on the notion of algebraic number

s, which includes all rational number

s, but also numbers such as the square root of 2

or

Complex number

s that are not algebraic are called transcendental

; for example, π and

complex numbers are transcendental. Using these notions, the Gelfond–Scheider theorem states that given two algebraic numbers

For any real number

This is a shorthand for saying that ln(

For example, the third approximation with yields 0.4167, about 0.011 more than . These terms approximate ln(

of this series

of sums. It is the Taylor series

of the natural logarithm at

For example, when

More efficient series

Another series is based on the Area hyperbolic tangent function:

for complex numbers

This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if

the logarithm of

The better the initial approximation

, which

converges quickly provided

A closely related method can be used to compute the logarithm of integers. From the above series, it follows that:

If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1).

):

Here

) and the square root of the product of two numbers (geometric mean

). Moreover,

Both the arithmetic-geometric mean and the constants π and ln(2) can be calculated with quickly converging series.

. For example, each chamber of the shell of a nautilus

is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral

. Benford's law

on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to self-similarity

. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions. The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.

Logarithmic scale

s are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(

, the Fenske equation

, or the Nernst equation

.

is a logarithmic unit of measurement. It is based on the common logarithm of ratio

s—10 times the common logarithm of a power

ratio or 20 times the common logarithm of a voltage

ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics

, and the absorbance of light in the fields of spectrometry

and optics

. The signal-to-noise ratio

describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels. In a similar vein, the peak signal-to-noise ratio

is commonly used to assess the quality of sound and image compression

methods using the logarithm.

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale

or the Richter scale. For example, a 5.0 earthquake releases 10 times and a 6.0 releases 100 times the energy of a 4.0. Another logarithmic scale is apparent magnitude

. It measures the brightness of stars logarithmically. Yet another example is pH

in chemistry

; pH is the negative of the common logarithm of the activity

of hydronium

ions (the form hydrogen

ion

s take in water). The activity of hydronium ions in neutral water is 10

Semilog (log-linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

s of the form

proportional to

Log-log graphs scale both axes logarithmically, which causes functions of the form

s.

Hick's law

proposes a logarithmic relation between the time individuals take for choosing an alternative and the number of choices they have. Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target. In psychophysics

, the Weber–Fechner law

proposes a logarithmic relationship between stimulus

and sensation such as the actual vs. the perceived weight of an item a person is carrying. (This "law", however, is less precise than more recent models, such as the Stevens' power law

.)

Psychological studies found that mathematically unsophisticated individuals tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 20 as 100 is to 200. Increasing mathematical understanding shifts this to a linear estimate (positioning 100 10x as far away).

: the law of large numbers

dictates that, for a fair coin

, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm

.

Logarithms also occur in log-normal distributions. When the logarithm of a random variable

has a normal distribution, the variable is said to have a log-normal distribution. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.

Logarithms are used for maximum-likelihood estimation of parametric statistical model

s. For such a model, the likelihood function

depends on at least one parameter

that needs to be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "

Benford's law

describes the occurrence of digits in many data set

s, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is

is a branch of computer science

that studies the performance

of algorithm

s (computer programs solving a certain problem). Logarithms are valuable for describing algorithms which divide a problem

into smaller ones, and join the solutions of the subproblems.

For example, to find a number in a sorted list, the binary search algorithm

checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log

algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to

. The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor, is usually disregarded in the analysis of algorithms under the standard uniform cost model.

A function

if

in no more than bit

s. In other words, the amount of memory needed to store

is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy

The sum is over all possible states

Lyapunov exponent

s use logarithms to gauge the degree of chaoticity of a dynamical system

. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic

in a deterministic way because small errors of measurement of the initial state will predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.

of fractal

s. Fractals are geometric objects that are self-similar

: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle

(pictured) can be covered by three copies of itself, each having sides half the original length. This causes the Hausdorff dimension

of this structure to be log(3)/log(2) ≈ 1.58. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.

. In equal temperament

, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch

, of the individual tones. For example, the note

has a frequency of 440 Hz

and

, as is the one between

(frequency 493 Hz). Accordingly, the frequency ratios agree:

Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the logarithm of the frequency

ratio, while the logarithm of the frequency ratio expresses the interval in cents

, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.

. For any integer

s less than or equal to

asserts that π(

in the sense that the ratio of π(

to the numbers of decimal digits of

offset logarithmic integral

function Li(

The Riemann hypothesis

, one of the oldest open mathematical conjecture

s, can be stated in terms of comparing π(

describing the number of distinct prime factor

s also involves the natural logarithm.

The logarithm of

,

This can be used to obtain Stirling's formula, an approximation of

s

are called

. Such a number can be visualized by a point in the complex plane

, as shown at the right. The polar form encodes a non-zero complex number

, that is, the distance

, and an angle between the

The argument φ is not uniquely specified by

s or 360 degrees to the argument φ corresponds to "winding" around the origin counter-clock-wise by an angle of 2π. The resulting complex number is again

Using trigonometric functions sine

and cosine, or the complex exponential, respectively,

This implies that the power of

φ is the principal argument Arg(

The illustration at the right depicts Log(

is the (multi-valued) inverse function of the matrix exponential

. Another example is the

. Both are defined via Taylor series analogous to the real case. In the context of differential geometry, the exponential map

maps the tangent space

at a point of a manifold

to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.

In the context of finite groups exponentiation is given by repeatedly multiplying one group element

is the integer

where

keys over unsecured information channels. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field

.

Further logarithm-like inverse functions include the

in computer science), the Lambert W function, and the logit

. They are the inverse functions of the double exponential function, tetration

, of , and of the logistic function

, respectively.

, the identity expresses a group isomorphism

between positive reals

under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure

(Lebesgue measure

)

and algebraic geometry

, differential form

s of the form

The polylogarithm is the function defined by

It is related to the natural logarithm by Li

**logarithm**of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if*x*=*b*^{y}, then*y*is the logarithm of*x*to base*b*, and is written log_{b}(*x*), so log_{10}(1000) = 3.Logarithms were introduced by John Napier

John Napier

John Napier of Merchiston – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer...

in the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily and rapidly, using slide rule

Slide rule

The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...

s and logarithm table

Common logarithm

The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...

s. These devices rely on the fact—important in its own right—that the logarithm of a product

Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...

is the sum

SUM

SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...

of the logarithms of the factors:

The present-day notion of logarithms comes from Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

, who connected them to the exponential function

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

in the 18th century.

The logarithm to base

*b*= 10 is called the common logarithmCommon logarithm

The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...

and has many applications in science and engineering. The natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

has the constant

*e*E (mathematical constant)

The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

(≈ 2.718) as its base; its use is widespread in pure mathematics

Pure mathematics

Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

, especially calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. The binary logarithm

Binary logarithm

In mathematics, the binary logarithm is the logarithm to the base 2. It is the inverse function of n ↦ 2n. The binary logarithm of n is the power to which the number 2 must be raised to obtain the value n. This makes the binary logarithm useful for anything involving powers of 2,...

uses base

*b*= 2 and is prominent in computer scienceComputer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

.

Logarithmic scale

Logarithmic scale

A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

s reduce wide-ranging quantities to smaller scopes. For example, the decibel

Decibel

The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

is a logarithmic unit quantifying sound pressure

Sound pressure

Sound pressure or acoustic pressure is the local pressure deviation from the ambient atmospheric pressure caused by a sound wave. Sound pressure can be measured using a microphone in air and a hydrophone in water...

and voltage ratios. In chemistry, pH

PH

In chemistry, pH is a measure of the acidity or basicity of an aqueous solution. Pure water is said to be neutral, with a pH close to 7.0 at . Solutions with a pH less than 7 are said to be acidic and solutions with a pH greater than 7 are basic or alkaline...

is a logarithmic measure for the acid

Acid

An acid is a substance which reacts with a base. Commonly, acids can be identified as tasting sour, reacting with metals such as calcium, and bases like sodium carbonate. Aqueous acids have a pH of less than 7, where an acid of lower pH is typically stronger, and turn blue litmus paper red...

ity of an aqueous solution

Aqueous solution

An aqueous solution is a solution in which the solvent is water. It is usually shown in chemical equations by appending aq to the relevant formula, such as NaCl. The word aqueous means pertaining to, related to, similar to, or dissolved in water...

. Logarithms are commonplace in scientific formula

Formula

In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

s, and in measurements of the complexity of algorithms

Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...

and of geometric objects called fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s. They describe musical intervals

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

, appear in formulas counting prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s, inform some models in psychophysics

Psychophysics

Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they effect. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual...

, and can aid in forensic accounting

Forensic accounting

Forensic accounting is the specialty practice area of accountancy that describes engagements that result from actual or anticipated disputes or litigation. "Forensic" means "suitable for use in a court of law", and it is to that standard and potential outcome that forensic accountants generally...

.

In the same way as the logarithm reverses exponentiation

Exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

, the complex logarithm

Complex logarithm

In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...

is the inverse function

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

of the exponential function applied to complex numbers. The discrete logarithm

Discrete logarithm

In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. In particular, an ordinary logarithm loga is a solution of the equation ax = b over the real or complex numbers...

is another variant; it has applications in public-key cryptography

Public-key cryptography

Public-key cryptography refers to a cryptographic system requiring two separate keys, one to lock or encrypt the plaintext, and one to unlock or decrypt the cyphertext. Neither key will do both functions. One of these keys is published or public and the other is kept private...

.

## Motivation and definition

The idea of*logarithms*is to reverse the operation of exponentiationExponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

, that is raising a number to a power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2:

It follows that the logarithm of 8 with respect to base 2 is 3.

### Exponentiation

The third power of some number*b*is the product of 3 factors of*b*. More generally, raising*b*to the power, where*n*is a natural numberNatural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

, is done by multiplying

*n*factors. The power of*b*is written*b*^{n}, so thatThe

*n*-th power of*b*,*b*^{n}, is defined whenever*b*is a positive number and*n*is a real numberReal number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

. For example,

*b*^{−1}is the reciprocalMultiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

of

*b*, that is, .### Definition

The logarithm of a number*x*with respect to base*b*is the exponent to which*b*has to be raised to yield*x*. In other words, the logarithm of*x*to base*b*is the solution*y*of the equationThe logarithm is denoted "log

_{b}(*x*)" . In the equation*y*= log_{b}(*x*), the value*y*, is the answer to the question "To what power must*b*be raised, in order to yield*x*?". For the logarithm to be defined, the base*b*must be a positive real number not equal to 1 and*x*must be a positive number.### Examples

For example, , since 16. Logarithms can also be negative:since

A third example: log

_{10}(150) is approximately 2.176, which lies between 2 and 3, just as 150 lies between and . Finally, for any base*b*, and , since and , respectively.## Logarithmic identities

Several important formulas, sometimes called*logarithmic identities*or*log laws*, relate logarithms to one another.### Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. Therefore, the logarithm of the power of a number is*p*times the logarithm of the number itself; the logarithm of a root is the logarithm of the number divided by*p*. The following table lists these identities with examples:Formula | Example | |
---|---|---|

product | ||

quotient | ||

power | ||

root |

### Change of base

The logarithm log_{b}(*x*) can be computed from the logarithms of*x*and*b*with respect to an arbitrary base*k*using the following formula:Typical scientific calculators calculate the logarithms to bases 10 and

*e*E (mathematical constant)

The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

. Logarithms with respect to any base

*b*can be determined using either of these two logarithms by the previous formula:Given a number

*x*and its logarithm log_{b}(*x*) to an unknown base*b*, the base is given by:## Particular bases

Among all choices for the base*b*, three are particularly common. These are*b*= 10,*b*=*e*E (mathematical constant)

The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

(the irrational

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

mathematical constant ≈ 2.71828), and

*b*= 2. In mathematical analysisMathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, the logarithm to base

*e*is widespread because of its particular analytical properties explained below. On the other hand, logarithms are easy to use for manual calculations in the decimalDecimal

The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

number system:

Thus, log

_{10}(*x*) is related to the number of decimal digits of a positive integer*x*: the number of digits is the smallest integerInteger

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

strictly bigger than log

_{10}(*x*). For example, log_{10}(1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430. The logarithm to base two is used in computer scienceComputer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, where the binary system

Binary numeral system

The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

is ubiquitous.

The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write log(

*x*) instead of log_{b}(*x*), when the intended base can be determined from the context. The notation^{b}log(*x*) also occurs. The "ISO notation" column lists designations suggested by the International Organization for StandardizationInternational Organization for Standardization

The International Organization for Standardization , widely known as ISO, is an international standard-setting body composed of representatives from various national standards organizations. Founded on February 23, 1947, the organization promulgates worldwide proprietary, industrial and commercial...

(ISO 31-11

ISO 31-11

ISO 31-11 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology...

).

Base b |
Name for log_{b}(x) |
ISO notation | Other notations | Used in |
---|---|---|---|---|

2 | binary logarithm Binary logarithm In mathematics, the binary logarithm is the logarithm to the base 2. It is the inverse function of n ↦ 2n. The binary logarithm of n is the power to which the number 2 must be raised to obtain the value n. This makes the binary logarithm useful for anything involving powers of 2,... |
lb(x) |
ld(x), log(x), lg(x) |
computer science, information theory Information theory Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and... , mathematics |

e |
natural logarithm Natural logarithm The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828... |
ln(x) |
log(x)(in mathematics and many programming language Programming language A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely.... s) |
mathematical analysis, physics, chemistry, statistics Statistics Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.... , economics Economics Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"... , and some engineering fields |

10 | common logarithm Common logarithm The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L... |
lg(x) |
log(x)(in engineering, biology, astronomy), |
various engineering Engineering Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of... fields (see decibel Decibel The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities... and see below), logarithm tables Mathematical table Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation... , handheld calculators Scientific calculator A scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathematics... |

### Predecessors

The Indian mathematician VirasenaVirasena

Āchārya Virasena was an 8th century Indian mathematician and Jain philosopher and scholar. He was a student of the Jain sage Elāchārya. He is also known to be a famous orator and an accomplished poet. His most reputed work is the Jain treatise Dhavala. Late Dr. Hiralal Jain places the completion of...

worked with the concept of

*ardhaccheda*: the number of times a number of the form 2^{n}could be halved. For exact powers of 2, this is the logarithm to that base, which is a whole number; for other numbers, it is undefined. He described relations such as the product formula and also introduced integer logarithms in base 3 (*trakacheda*) and base 4 (*caturthacheda*). Michael StifelMichael Stifel

Michael Stifel or Styfel was a German monk and mathematician. He was an Augustinian who became an early supporter of Martin Luther. Stifel was later appointed professor of mathematics at Jena University...

published

*Arithmetica integra*in NurembergNuremberg

Nuremberg[p] is a city in the German state of Bavaria, in the administrative region of Middle Franconia. Situated on the Pegnitz river and the Rhine–Main–Danube Canal, it is located about north of Munich and is Franconia's largest city. The population is 505,664...

in 1544 which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.

In the 16th and early 17th centuries an algorithm called prosthaphaeresis

Prosthaphaeresis

Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally-applicable way of approximating...

was used to approximate multiplication and division. This used the trigonometrical identity

or similar or convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.

The Babylonians

Babylonian mathematics

Babylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

sometime in 2000–1600 BC invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of squares. However it could not be used for division without an additional table of reciprocals. This method was used to simplify the accurate multiplication of large numbers till superseded by the use of computers.

### From Napier to Euler

The method of logarithms was publicly propounded by John NapierJohn Napier

John Napier of Merchiston – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer...

in 1614, in a book entitled

*Mirifici Logarithmorum Canonis Descriptio*(*Description of the Wonderful Rule of Logarithms*). Joost BürgiJoost Bürgi

Jost Bürgi, or Joost, or Jobst Bürgi , active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomical instruments and a mathematician....

independently invented logarithms but published six years after Napier.

By repeated subtractions Napier calculated for

*L*ranging from 1 to 100. The result for*L*=100 is approximately 0.99999 = 1 − 10^{−5}. Napier then calculated the products of these numbers with for*L*from 1 to 50, and did similarly with and . These computations, which occupied 20 years, allowed him to give, for any number*N*from 5 to 10 million, the number*L*that solves the equationNapier first called

*L*an "artificial number", but later introduced the word*"logarithm"*to mean a number that indicates a ratio: (*logos*

) meaning proportion, and (Logos

' is an important term in philosophy, psychology, rhetoric and religion. Originally a word meaning "a ground", "a plea", "an opinion", "an expectation", "word," "speech," "account," "reason," it became a technical term in philosophy, beginning with Heraclitus ' is an important term in...

*arithmos*) meaning number. In modern notation, the relation to natural logarithms is:where the very close approximation corresponds to the observation that

The invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri

Bonaventura Cavalieri

Bonaventura Francesco Cavalieri was an Italian mathematician. He is known for his work on the problems of optics and motion, work on the precursors of infinitesimal calculus, and the introduction of logarithms to Italy...

(Italy), Edmund Wingate

Edmund Wingate

Edmund Wingate was an English mathematical and legal writer, one of the first to publish in the 1620s on the principle of the slide rule, and later the author of some popular expository works...

(France), Xue Fengzuo (China), and

Johannes Kepler

Johannes Kepler

Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

's

*Chilias logarithmorum*(Germany) helped spread the concept further.In 1647 Grégoire de Saint-Vincent

Grégoire de Saint-Vincent

Grégoire de Saint-Vincent , a Jesuit, was a mathematician who discovered that the area under a rectangular hyperbola is the same over [a,b] as over [c,d] when a/b = c/d...

related logarithms to the quadrature of the hyperbola, by pointing out that the area

*f*(*t*) under the hyperbola from to satisfiesThe natural logarithm was first described by Nicholas Mercator

Nicholas Mercator

Nicholas Mercator , also known by his Germanic name Kauffmann, was a 17th-century mathematician....

in his work

*Logarithmotechnia*published in 1668, although the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm. Around 1730, Leonhard EulerLeonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

defined the exponential function and the natural logarithm by

Euler also showed that the two functions are inverse to one another.

### Logarithm tables, slide rules, and historical applications

By simplifying difficult calculations, logarithms contributed to the advance of science, and especially of astronomyAstronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

. They were critical to advances in surveying

Surveying

See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

, celestial navigation

Celestial navigation

Celestial navigation, also known as astronavigation, is a position fixing technique that has evolved over several thousand years to help sailors cross oceans without having to rely on estimated calculations, or dead reckoning, to know their position...

, and other domains. Pierre-Simon Laplace

Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

called logarithms

[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.

A key tool that enabled the practical use of logarithms before calculators and computers was the

*table of logarithms*. The first such table was compiled by Henry BriggsHenry Briggs (mathematician)

Henry Briggs was an English mathematician notable for changing the original logarithms invented by John Napier into common logarithms, which are sometimes known as Briggsian logarithms in his honour....

in 1617, immediately after Napier's invention. Subsequently, tables with increasing scope and precision were written. These tables listed the values of log

_{b}(*x*) and*b*^{x}for any number*x*in a certain range, at a certain precision, for a certain base*b*(usually*b*= 10). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits. As the function is the inverse function of log_{b}(*x*), it has been called the antilogarithm. The product and quotient of two positive numbers*c*and*d*were routinely calculated as the sum and difference of their logarithms. The product*cd*or quotient*c*/*d*came from looking up the antilogarithm of the sum or difference, also via the same table:and

For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis

Prosthaphaeresis

Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally-applicable way of approximating...

, which relies on trigonometric identities. Calculations of powers and roots

Nth root

In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

are reduced to multiplications or divisions and look-ups by

and

Many logarithm tables give logarithms by separately providing the characteristic and mantissa

Common logarithm

of

*x*, that is to say, the integer part and the fractional partFractional part

All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...

of log

_{10}(*x*). The characteristic of is one plus the characteristic of*x*, and their significands are the same. This extends the scope of logarithm tables: given a table listing log_{10}(*x*) for all integers*x*ranging from 1 to 1000, the logarithm of 3542 is approximated byAnother critical application was the slide rule

Slide rule

The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...

, a pair of logarithmically divided scales used for calculation, as illustrated here:

The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred

William Oughtred

William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...

enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of less precision, much faster computation than techniques based on tables.

## Analytic properties

A deeper study of logarithms requires the concept of a*function*

. A function is a rule that, given one number, produces another number. An example is the function producing the power ofFunction (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

*b*from any real number*x*, where the base (or radixRadix

In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...

)

*b*is a fixed number. This function is written### Logarithmic function

To justify the definition of logarithms, it is necessary to show that the equationhas a solution

*x*and that this solution is unique, provided that*y*is positive and that*b*is positive and unequal to 1. A proof of that fact requires the intermediate value theoremIntermediate value theorem

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....

from elementary calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. This theorem states that a continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

which produces two values

*m*and*n*also produces any value that lies between*m*and*n*. A function is*continuous*if it does not "jump", that is, if its graph can be drawn without lifting the pen.This property can be shown to hold for the function

*f*(*x*) =*b*^{x}. Because*f*takes arbitrarily large and arbitrarily small positive values, any number lies between*f*(*x*_{0}) and*f*(*x*_{1}) for suitable*x*_{0}and*x*_{1}. Hence, the intermediate value theorem ensures that the equation*f*(*x*) =*y*has a solution. Moreover, there is only one solution to this equation, because the function*f*is strictly increasingMonotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

(for ), or strictly decreasing (for ).

The unique solution

*x*is the logarithm of*y*to base*b*, log_{b}(*y*). The function which assigns to*y*its logarithm is called*logarithm function*or*logarithmic function*(or just*logarithm*).### Inverse function

The formula for the logarithm of a power says in particular that for any number*x*,In prose, taking the power of

*b*and then the logarithm gives back*x*. Conversely, given a positive number*y*, the formulasays that first taking the logarithm and then exponentiating gives back

*y*. Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base*b*is the*inverse function*

of .Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

Inverse functions are closely related to the original functions. Their graphs

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

correspond to each other upon exchanging the

*x*- and the*y*-coordinates (or upon reflection at the diagonal line*x*=*y*), as shown at the right: a point (*t*,*u*=*b*^{t}) on the graph of*f*yields a point (*u*,*t*= log_{b}*u*) on the graph of the logarithm and vice versa. As a consequence, log_{b}(*x*) diverges to infinity (gets bigger than any given number) if*x*grows to infinity, provided that*b*is greater than one. In that case, log_{b}(*x*) is an increasing function. For , log_{b}(*x*) tends to minus infinity instead. When*x*approaches zero, log_{b}(*x*) goes to minus infinity for (plus infinity for , respectively).### Derivative and antiderivative

Analytic properties of functions pass to their inverses. Thus, as*f*(*x*) =*b*^{x}is a continuous and differentiable functionDifferentiable function

In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...

, so is log

_{b}(*y*). Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivativeDerivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

of

*f*(*x*) evaluates to ln(*b*)*b*^{x}by the properties of the exponential functionExponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

, the chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

implies that the derivative of log

_{b}(*x*) is given byThat is, the slope

Slope

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

of the tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

touching the graph of the logarithm at the point equals . In particular, the derivative of ln(

*x*) is 1/*x*, which implies that the antiderivativeAntiderivative

In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

of 1/

*x*is . The derivative with a generalised functional argument*f*(*x*) isThe quotient at the right hand side is called the logarithmic derivative

Logarithmic derivative

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....

of

*f*. Computing*f*' (*x*) by means of the derivative of ln(*f*(*x*)) is known as logarithmic differentiation. The antiderivative of the natural logarithm ln(*x*) is:Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.

### Integral representation of the natural logarithm

The natural logarithm of*t*agrees with the integralIntegral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of 1/

*x**dx*from 1 to*t*:In other words, ln(

*t*) equals the area between the*x*axis and the graph of the function 1/*x*, ranging from*x*= 1 to*x*=*t*(figure at the right). This is a consequence of the fundamental theorem of calculusFundamental theorem of calculus

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

and the fact that derivative of ln(

*x*) is 1/*x*. The right hand side of this equation can serve as a definition of the natural logarithm. Product and power logarithm formulas can be derived from this definition. For example, the product formula ln(*tu*) = ln(*t*) + ln(*u*) is deduced as:The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (

*w*=*x*/*t*). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor*t*and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function*f*(*x*) = 1/*x*again. Therefore, the left hand blue area, which is the integral of*f*(*x*) from*t*to*tu*is the same as the integral from*1*to*u*. This justifies the equality (2) with a more geometric proof.The power formula ln(

*t*^{r}) =*r*ln(*t*) may be derived in a similar way:The second equality uses a change of variables (integration by substitution

Integration by substitution

In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...

),

*w*:=*x*^{1/r}.The sum over the reciprocals of natural numbers,

is called the harmonic series

Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

. It is closely tied to the natural logarithm: as

*n*tends to infinityInfinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

, the difference,

converges

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

(i.e., gets arbitrarily close) to a number known as the Euler–Mascheroni constant

Euler–Mascheroni constant

The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

. This relation aids in analyzing the performance of algorithms such as quicksort.

## Calculation

Logarithms are easy to compute in some cases, such as log_{10}(1,000) = 3. In general, logarithms can be calculated using power series or the arithmetic-geometric mean or retrieved from a precalculated logarithm table that provides a fixed precision. Moreover, the binary logarithm algorithm calculates lb(*x*) recursivelyRecursion

Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

based on repeated squarings of

*x*, taking advantage of the relationNewton's method

Newton's method

In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...

, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently. Using look-up tables, CORDIC

CORDIC

CORDIC is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions...

-like methods can be used to compute logarithms if the only available operations are addition and bit shifts

Arithmetic shift

In computer programming, an arithmetic shift is a shift operator, sometimes known as a signed shift . For binary numbers it is a bitwise operation that shifts all of the bits of its operand; every bit in the operand is simply moved a given number of bit positions, and the vacant bit-positions are...

.

From a theoretical point of view, the Gelfond–Schneider theorem

Gelfond–Schneider theorem

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...

asserts that logarithms usually take "difficult" values. The formal statement relies on the notion of algebraic number

Algebraic number

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

s, which includes all rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s, but also numbers such as the square root of 2

Square root of 2

The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...

or

Complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s that are not algebraic are called transcendental

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

; for example, π and

*e*are such numbers. Almost allAlmost all

In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

complex numbers are transcendental. Using these notions, the Gelfond–Scheider theorem states that given two algebraic numbers

*a*and*b*, log_{b}(*a*) is either a transcendental number or a rational number*p*/*q*(in which case*a*^{q}=*b*^{p}, so*a*and*b*were closely related to begin with).### Power series

Taylor seriesFor any real number

*z*that satisfies , the following formula holds:This is a shorthand for saying that ln(

*z*) can be approximated to a more and more accurate value by the following expressions:For example, the third approximation with yields 0.4167, about 0.011 more than . These terms approximate ln(

*z*) with arbitrary precision, provided the number of summands is big enough. In elementary calculus, ln(*z*) is therefore called the*limit*Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

of this series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

of sums. It is the Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

of the natural logarithm at

*z*= 1. The Taylor series to ln*z*provides a particularly useful approximation to ln(1+*z*) when*z*is small,*|z| << 1*, since thenFor example, when

*z*= 0.1, the first-order approximation gives ln(1.1) ≈ 0.1, less than 5% off the correct value 0.0953.More efficient series

Another series is based on the Area hyperbolic tangent function:

for complex numbers

*z*with positive real part. Using the Sigma notation this is also written asThis series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if

*z*is close to 1. For example, for*z*= 1.5, the first three terms of the second series approximate ln(1.5) with an error of about . The quick convergence for*z*close to 1 can be taken advantage of in the following way: given a low-accuracy approximation and puttingthe logarithm of

*z*is:The better the initial approximation

*y*is, the closer*A*is to 1, so its logarithm can be calculated efficiently.*A*can be calculated using the exponential seriesExponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

, which

converges quickly provided

*y*is not too large. Calculating the logarithm of larger*z*can be reduced to smaller values of*z*by writing , so that .A closely related method can be used to compute the logarithm of integers. From the above series, it follows that:

If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1).

### Arithmetic-geometric mean approximation

The arithmetic-geometric mean yields high precision approximations of the natural logarithm. ln(*x*) is approximated to a precision of 2^{−p}(or*p*precise bits) by the following formula (from Carl Friedrich GaussCarl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

):

Here

*M*denotes the arithmetic-geometric mean. It is obtained by repeatedly calculating the average (arithmetic meanArithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

) and the square root of the product of two numbers (geometric mean

Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

). Moreover,

*m*is chosen such thatBoth the arithmetic-geometric mean and the constants π and ln(2) can be calculated with quickly converging series.

## Applications

Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invarianceScale invariance

In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...

. For example, each chamber of the shell of a nautilus

Nautilus

Nautilus is the common name of marine creatures of cephalopod family Nautilidae, the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina. It comprises six living species in two genera, the type of which is the genus Nautilus...

is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral

Logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous...

. Benford's law

Benford's law

Benford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is distributed in a specific, non-uniform way...

on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to self-similarity

Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions. The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.

Logarithmic scale

Logarithmic scale

A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

s are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(

*x*) grows very slowly for large*x*, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equationTsiolkovsky rocket equation

The Tsiolkovsky rocket equation, or ideal rocket equation is an equation that is useful for considering vehicles that follow the basic principle of a rocket: where a device that can apply acceleration to itself by expelling part of its mass with high speed and moving due to the conservation of...

, the Fenske equation

Fenske equation

The Fenske equation in continuous fractional distillation is an equation used for calculating the minimum number of theoretical plates required for the separation of a binary feed stream by a fractionation column that is being operated at total reflux .The equation was derived by Merrell Fenske in...

, or the Nernst equation

Nernst equation

In electrochemistry, the Nernst equation is an equation that can be used to determine the equilibrium reduction potential of a half-cell in an electrochemical cell. It can also be used to determine the total voltage for a full electrochemical cell...

.

### Logarithmic scale

Scientific quantities are often expressed as logarithms of other quantities, using a*logarithmic scale*. For example, the decibelDecibel

The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

is a logarithmic unit of measurement. It is based on the common logarithm of ratio

Ratio

In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

s—10 times the common logarithm of a power

Power (physics)

In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

ratio or 20 times the common logarithm of a voltage

Voltage

Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics

Acoustics

Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

, and the absorbance of light in the fields of spectrometry

Spectrometer

A spectrometer is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify materials. The variable measured is most often the light's intensity but could also, for instance, be the polarization...

and optics

Optics

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

. The signal-to-noise ratio

Signal-to-noise ratio

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...

describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels. In a similar vein, the peak signal-to-noise ratio

Peak signal-to-noise ratio

The phrase peak signal-to-noise ratio, often abbreviated PSNR, is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation...

is commonly used to assess the quality of sound and image compression

Image compression

The objective of image compression is to reduce irrelevance and redundancy of the image data in order to be able to store or transmit data in an efficient form.- Lossy and lossless compression :...

methods using the logarithm.

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale

Moment magnitude scale

The moment magnitude scale is used by seismologists to measure the size of earthquakes in terms of the energy released. The magnitude is based on the seismic moment of the earthquake, which is equal to the rigidity of the Earth multiplied by the average amount of slip on the fault and the size of...

or the Richter scale. For example, a 5.0 earthquake releases 10 times and a 6.0 releases 100 times the energy of a 4.0. Another logarithmic scale is apparent magnitude

Apparent magnitude

The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...

. It measures the brightness of stars logarithmically. Yet another example is pH

PH

In chemistry, pH is a measure of the acidity or basicity of an aqueous solution. Pure water is said to be neutral, with a pH close to 7.0 at . Solutions with a pH less than 7 are said to be acidic and solutions with a pH greater than 7 are basic or alkaline...

in chemistry

Chemistry

Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

; pH is the negative of the common logarithm of the activity

Activity (chemistry)

In chemical thermodynamics, activity is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.By convention, activity...

of hydronium

Hydronium

In chemistry, a hydronium ion is the cation , a type of oxonium ion produced by protonation of water. This cation is often used to represent the nature of the proton in aqueous solution, where the proton is highly solvated...

ions (the form hydrogen

Hydrogen

Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

ion

Ion

An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

s take in water). The activity of hydronium ions in neutral water is 10

^{−7}mol·L^{−1}, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10^{4}of the activity, that is, vinegar's hydronium ion activity is about 10^{−3}mol·L^{−1}.Semilog (log-linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential function

Exponential function

s of the form

*f*(*x*) =*a*·*b*^{x}appear as straight lines with slopeSlope

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

proportional to

*b*.Log-log graphs scale both axes logarithmically, which causes functions of the form

*f*(*x*) =*a*·*x*^{k}to be depicted as straight lines with slope proportional to the exponent*k*. This is applied in visualizing and analyzing power lawPower law

A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

s.

### Psychology

Logarithms occur in several laws describing human perception:Hick's law

Hick's law

Hick's Law, named after British psychologist William Edmund Hick, or the Hick–Hyman Law , describes the time it takes for a person to make a decision as a result of the possible choices he or she has. The Hick-Hyman Law assesses cognitive information capacity in choice reaction experiments...

proposes a logarithmic relation between the time individuals take for choosing an alternative and the number of choices they have. Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target. In psychophysics

Psychophysics

Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they effect. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual...

, the Weber–Fechner law

Weber–Fechner law

The Weber–Fechner law is a confusing term, because it combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law. Fechner himself added confusion to the literature by calling his own law Weber's law...

proposes a logarithmic relationship between stimulus

Stimulus (psychology)

In psychology, stimuli are energy patterns which are registered by the senses. In behaviorism and related stimulus–response theories, stimuli constitute the basis for behavior, whereas in perceptual psychology they constitute the basis for perception.In the second half of the 19th century, the...

and sensation such as the actual vs. the perceived weight of an item a person is carrying. (This "law", however, is less precise than more recent models, such as the Stevens' power law

Stevens' power law

Stevens' power law is a proposed relationship between the magnitude of a physical stimulus and its perceived intensity or strength. It is often considered to supersede the Weber–Fechner law on the basis that it describes a wider range of sensations, although critics argue that the validity of the...

.)

Psychological studies found that mathematically unsophisticated individuals tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 20 as 100 is to 200. Increasing mathematical understanding shifts this to a linear estimate (positioning 100 10x as far away).

### Probability theory and statistics

Logarithms arise in probability theoryProbability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

: the law of large numbers

Law of large numbers

In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

dictates that, for a fair coin

Fair coin

In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin...

, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm

Law of the iterated logarithm

In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Y. Khinchin . Another statement was given by A.N...

.

Logarithms also occur in log-normal distributions. When the logarithm of a random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

has a normal distribution, the variable is said to have a log-normal distribution. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.

Logarithms are used for maximum-likelihood estimation of parametric statistical model

Statistical model

A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...

s. For such a model, the likelihood function

Likelihood function

In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...

depends on at least one parameter

Parametric model

In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters...

that needs to be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "

*log likelihood*"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables.Benford's law

Benford's law

Benford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is distributed in a specific, non-uniform way...

describes the occurrence of digits in many data set

Data set

A data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...

s, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is

*d*(from 1 to 9) equals log_{10}(*d*+ 1) − log_{10}(*d*),*regardless*of the unit of measurement. Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.### Computational complexity

Analysis of algorithmsAnalysis of algorithms

To analyze an algorithm is to determine the amount of resources necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length...

is a branch of computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

that studies the performance

Time complexity

In computer science, the time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the size of the input to the problem. The time complexity of an algorithm is commonly expressed using big O notation, which suppresses multiplicative constants and...

of algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

s (computer programs solving a certain problem). Logarithms are valuable for describing algorithms which divide a problem

Divide and conquer algorithm

In computer science, divide and conquer is an important algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same type, until these become simple enough to be solved directly...

into smaller ones, and join the solutions of the subproblems.

For example, to find a number in a sorted list, the binary search algorithm

Binary search algorithm

In computer science, a binary search or half-interval search algorithm finds the position of a specified value within a sorted array. At each stage, the algorithm compares the input key value with the key value of the middle element of the array. If the keys match, then a matching element has been...

checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log

_{2}(*N*) comparisons, where*N*is the list's length. Similarly, the merge sortMerge sort

Merge sort is an O comparison-based sorting algorithm. Most implementations produce a stable sort, meaning that the implementation preserves the input order of equal elements in the sorted output. It is a divide and conquer algorithm...

algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to

Big O notation

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

. The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor, is usually disregarded in the analysis of algorithms under the standard uniform cost model.

A function

*f*(*x*) is said to grow logarithmicallyLogarithmic growth

In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log . Note that any logarithm base can be used, since one can be converted to another by a fixed constant...

if

*f*(*x*) is (exactly or approximately) proportional to the logarithm of*x*. (Biological descriptions of organism growth, however, use this term for an exponential function.) For example, any natural numberNatural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

*N*can be represented in binary formBinary numeral system

The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

in no more than bit

Bit

A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...

s. In other words, the amount of memory needed to store

*N*grows logarithmically with*N*.### Entropy and chaos

EntropyEntropy

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy

*S*of some physical system is defined asThe sum is over all possible states

*i*of the system in question, such as the positions of gas particles in a container. Moreover,*p*_{i}is the probability that the state*i*is attained and*k*is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of*N*possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log_{2}(*N*) bits.Lyapunov exponent

Lyapunov exponent

In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...

s use logarithms to gauge the degree of chaoticity of a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

in a deterministic way because small errors of measurement of the initial state will predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.

### Fractals

Logarithms occur in definitions of the dimensionFractal dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

of fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s. Fractals are geometric objects that are self-similar

Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle

Sierpinski triangle

The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...

(pictured) can be covered by three copies of itself, each having sides half the original length. This causes the Hausdorff dimension

Hausdorff dimension

thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...

of this structure to be log(3)/log(2) ≈ 1.58. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.

### Music

Logarithms are related to musical tones and intervalsInterval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. In equal temperament

Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch

Pitch (music)

Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

, of the individual tones. For example, the note

*A*A (musical note)

La or A is the sixth note of the solfège. "A" is generally used as a standard for tuning. When the orchestra tunes, the oboe plays an "A" and the rest of the instruments tune to match that pitch. Every string instrument in the orchestra has an A string, from which each player can tune the rest of...

has a frequency of 440 Hz

Hertz

The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

and

*B-flat*has a frequency of 466 Hz. The interval between*A*and*B-flat*is a semitoneSemitone

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

, as is the one between

*B-flat*and*B*B (musical note)

B, also known as H, Si or Ti, is the seventh note of the solfège. It lies a chromatic semitone below C and is thus the enharmonic equivalent of C-flat....

(frequency 493 Hz). Accordingly, the frequency ratios agree:

Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the logarithm of the frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ratio, while the logarithm of the frequency ratio expresses the interval in cents

Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.

Interval(the two tones are played at the same time) |
1/12 tone | Semitone Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
Just major third | Major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
Tritone Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
Octave Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"... |

Frequency ratio r |
||||||

Corresponding number of semitones |
||||||

Corresponding number of cents |

### Number theory

Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theoryNumber theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. For any integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

*x*, the quantity of prime numberPrime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s less than or equal to

*x*is denoted π(*x*). The prime number theoremPrime number theorem

In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

asserts that π(

*x*) is approximately given byin the sense that the ratio of π(

*x*) and that fraction approaches 1 when*x*tends to infinity. As a consequence, the probability that a randomly chosen number between 1 and*x*is prime is inversely proportionalProportionality (mathematics)

In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

to the numbers of decimal digits of

*x*. A far better estimate of π(*x*) is given by theoffset logarithmic integral

Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li is a special function. It occurs in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.-Integral...

function Li(

*x*), defined byThe Riemann hypothesis

Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

, one of the oldest open mathematical conjecture

Conjecture

A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

s, can be stated in terms of comparing π(

*x*) and Li(*x*). The Erdős–Kac theoremErdos–Kac theorem

In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω is the number of distinct prime factors of n, then, loosely speaking, the probability distribution ofis the standard normal...

describing the number of distinct prime factor

Prime factor

In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...

s also involves the natural logarithm.

The logarithm of

*n*factorialFactorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

,

*n*! = 1 · 2 · ... ·*n*, is given byThis can be used to obtain Stirling's formula, an approximation of

*n*! for large*n*.### Complex logarithm

The complex numberComplex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s

*a*solving the equationare called

*complex logarithms*. Here,*z*is a complex number. A complex number is commonly represented as*z = x + iy*, where*x*and*y*are real numbers and*i*is the imaginary unitImaginary unit

In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. Such a number can be visualized by a point in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, as shown at the right. The polar form encodes a non-zero complex number

*z*by its absolute valueAbsolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

, that is, the distance

*r*to the originOrigin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

, and an angle between the

*x*axis and the line passing through the origin and*z*. This angle is called argument of*z*. The absolute value*r*of*z*isThe argument φ is not uniquely specified by

*z*: φ' = φ + 2π is also an argument of*z*because adding 2π radianRadian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s or 360 degrees to the argument φ corresponds to "winding" around the origin counter-clock-wise by an angle of 2π. The resulting complex number is again

*z*, as illustrated at the right. However, exactly one argument φ satisfies and . It is called the*principal argument*, denoted Arg(*z*), with a capital*A*. (An alternative normalization is .)Using trigonometric functions sine

Sine

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

and cosine, or the complex exponential, respectively,

*r*and φ are such that the following identities hold:This implies that the power of

*e*equals*z*, whereφ is the principal argument Arg(

*z*) and*n*is an arbitrary integer. Any such*a*is called a complex logarithm of*z*. There are infinitely many of them, in contrast to the uniquely defined real logarithm. If*n*= 0,*a*is called the*principal value*of the logarithm, denoted Log(*z*). The principal argument of any positive real number*x*is 0; hence Log(*x*) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do*not*generalize to the principal value of the complex logarithm.The illustration at the right depicts Log(

*z*). The discontinuity, that is, the jump in the hue at the negative part of the*x*- or real axis, is caused by the jump of the principal argument there. This locus is called a branch cut. This behavior can only be circumvented by dropping the range restriction on φ. Then the argument of*z*and, consequently, its logarithm become multi-valued functions.### Inverses of other exponential functions

Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrixLogarithm of a matrix

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and...

is the (multi-valued) inverse function of the matrix exponential

Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....

. Another example is the

*p*-adic logarithm, the inverse function of the*p*-adic exponentialP-adic exponential function

In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers...

. Both are defined via Taylor series analogous to the real case. In the context of differential geometry, the exponential map

Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....

maps the tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

at a point of a manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.

In the context of finite groups exponentiation is given by repeatedly multiplying one group element

*b*with itself. The discrete logarithmDiscrete logarithm

In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. In particular, an ordinary logarithm loga is a solution of the equation ax = b over the real or complex numbers...

is the integer

*n*solving the equationwhere

*x*is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographicCryptography

Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

keys over unsecured information channels. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

.

Further logarithm-like inverse functions include the

*double logarithm*ln(ln(*x*)), the*super- or hyper-4-logarithm*

(a slight variation of which is called iterated logarithmSuper-logarithm

In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, super-roots and super-logarithms...

Iterated logarithm

In computer science, the iterated logarithm of n, written n , is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1...

in computer science), the Lambert W function, and the logit

Logit

The logit function is the inverse of the sigmoidal "logistic" function used in mathematics, especially in statistics.Log-odds and logit are synonyms.-Definition:The logit of a number p between 0 and 1 is given by the formula:...

. They are the inverse functions of the double exponential function, tetration

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

, of , and of the logistic function

Logistic function

A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...

, respectively.

### Related concepts

From the perspective of pure mathematicsPure mathematics

Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

, the identity expresses a group isomorphism

Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...

between positive reals

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure

Haar measure

In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....

(Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

)

*dx*on the reals corresponds to the Haar measure*dx*/*x*on the positive reals. In complex analysisComplex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

and algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, differential form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s of the form

*df*/*f*are known as forms with logarithmic poles.The polylogarithm is the function defined by

It is related to the natural logarithm by Li

_{1}(*z*) = −ln(1 −*z*). Moreover, Li_{s}(1) equals the Riemann zeta function ζ(*s*).