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Logarithm

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Inverse of the exponential function, which maps products to sums

Plots of logarithm functions, with three commonly used bases. The special points logbb = 1 are indicated by dotted lines, and all curves intersect in logb 1 = 0.

In

mathematics

, the logarithm is the

inverse function

to

exponentiation

. That means the logarithm of a given number x is the

exponent

to which another fixed number, the

base

 b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the “logarithm base 10” of 1000 is 3, or log10(1000) = 3. The logarithm of x to base b is denoted as logb(x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in

big O notation

.

More generally, exponentiation allows any positive

real number

as base to be raised to any real power, always producing a positive result, so logb(x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

logb⁡(x)=y {displaystyle log _{b}(x)=y } exactly if  by=x {displaystyle b^{y}=x } and  x>0{displaystyle x>0} and  b>0{displaystyle b>0} and  b≠1{displaystyle bneq 1}.

For example, log2 64 = 6, as 26 = 64.

The logarithm base 10 (that is b = 10) is called the decimal or

common logarithm

and is commonly used in science and engineering. The

natural logarithm

has the

number e

 (that is b ≈ 2.718) as its base; its use is widespread in mathematics and

physics

, because of its simpler

integral

and

derivative

. The

binary logarithm

uses base 2 (that is b = 2) and is frequently used in

computer science

. Logarithms are examples of

concave functions

.

[1]

Logarithms were introduced by

John Napier

in 1614 as a means of simplifying calculations.

[2]

They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using

logarithm tables

, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a

product

is the

sum

of the logarithms of the factors:

logb⁡(xy)=logb⁡x+logb⁡y,{displaystyle log _{b}(xy)=log _{b}x+log _{b}y,,}

provided that b, x and y are all positive and b ≠ 1. The

slide rule

, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from

Leonhard Euler

, who connected them to the

exponential function

in the 18th century, and who also introduced the letter e as the base of natural logarithms.

[3]

Logarithmic scales

reduce wide-ranging quantities to tiny scopes. For example, the

decibel

(dB) is a

unit

used to express

ratio as logarithms

, mostly for signal power and amplitude (of which

sound pressure

is a common example). In chemistry,

pH

is a logarithmic measure for the

acidity

of an

aqueous solution

. Logarithms are commonplace in scientific

formulae

, and in measurements of the

complexity of algorithms

and of geometric objects called

fractals

. They help to describe

frequency

ratios of

musical intervals

, appear in formulas counting

prime numbers

or

approximating

factorials

, 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, whether applied to

real numbers

or

complex numbers

. The modular

discrete logarithm

is another variant; it has uses in

public-key cryptography

.

Motivation and definition[

edit

]

Graph showing a logarithmic curve, crossing the x-axis at x= 1 and approaching minus infinity along the y-axis.

The

graph

of the logarithm base 2 crosses the

x-axis

at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) = 3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but

does not meet it

.

Addition

,

multiplication

, and

exponentiation

are three of the most fundamental arithmetic operations. Addition, the simplest of these, is undone by subtraction: when you add 5 to x to get x + 5, to reverse this operation you need to subtract 5 from x + 5. Multiplication, the next-simplest operation, is undone by

division

: if you multiply x by 5 to get 5x, you then can divide 5x by 5 to return to the original expression x. Logarithms also undo a fundamental arithmetic operation, exponentiation. Exponentiation is when you raise a number to a certain power. For example, raising 2 to the power 3 equals 8:

23=2×2=8{displaystyle 2^{3}=2times 2times 2=8}

The general case is when you raise a number b to the power of y to get x:

by=x{displaystyle b^{y}=x}

The number b is referred to as the base of this expression. The base is the number that is raised to a particular power—in the above example, the base of the expression 23=8{displaystyle 2^{3}=8} is 2. It is easy to make the base the subject of the expression: all you have to do is take the y-th root of both sides. This gives:

b=xy{displaystyle b={sqrt[{y}]{x}}}

It is less easy to make y the subject of the expression. Logarithms allow us to do this:

y=logb⁡x{displaystyle y=log _{b}x}

This expression means that y is equal to the power that you would raise b to, to get x. This operation undoes exponentiation because the logarithm of x tells you the exponent that the base has been raised to.

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Exponentiation[

edit

]

This subsection contains a short overview of the exponentiation operation, which is fundamental to understanding logarithms. Raising b to the n-th power, where n is a

natural number

, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that

bn=b××b⏟n factors{displaystyle b^{n}=underbrace {btimes btimes cdots times b} _{n{text{ factors}}}}

Exponentiation may be extended to by, where b is a positive number and the exponent y is any

real number

.

[4]

For example, b−1 is the

reciprocal

of b, that is, 1/b. Raising b to the power 1/2 is the

square root

of b.

More generally, raising b to a

rational

power p/q, where p and q are integers, is given by

bp/q=bpq,{displaystyle b^{p/q}={sqrt[{q}]{b^{p}}},}

the q-th root of bp{displaystyle b^{p}!!}.

Finally, any

irrational number

(a real number which is not rational) y can be approximated to arbitrary precision by rational numbers. This can be used to compute the y-th power of b: for example 2≈1.414…{displaystyle {sqrt {2}}approx 1.414…} and b2{displaystyle b^{sqrt {2}}} is increasingly well approximated by b1,b1.4,b1.41,b1.414,…{displaystyle b^{1},b^{1.4},b^{1.41},b^{1.414},…}. A more detailed explanation, as well as the formula bm + n = bm · bn is contained in the article on

exponentiation

.

Definition[

edit

]

The logarithm of a positive real number x with respect to base b

[nb 1]

is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the solution y to the equation

[5]

by=x.{displaystyle b^{y}=x.}

The logarithm is denoted “logb x” (pronounced as “the logarithm of x to base b” or “the base-b logarithm of x” or (most commonly) “the log, base b, of x“).

In the equation y = logb x, the value y is the answer to the question “To what power must b be raised, in order to yield x?”.

Examples[

edit

]

  • log2 16 = 4 , since 24 = 2 ×2 × 2 × 2 = 16.
  • Logarithms can also be negative: log212=−1{displaystyle quad log _{2}!{frac {1}{2}}=-1quad } since 2−1=121=12.{displaystyle quad 2^{-1}={frac {1}{2^{1}}}={frac {1}{2}}.}
  • log10 150 is approximately 2.176, which lies between 2 and 3, just as 150 lies between 102 = 100 and 103 = 1000.
  • For any base b, logbb = 1 and logb 1 = 0, since b1 = b and b0 = 1, respectively.

Logarithmic identities[

edit

]

Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.

[6]

Product, quotient, power, and root[

edit

]

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. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions x=blogb⁡x{displaystyle x=b^{log _{b}x}} or y=blogb⁡y{displaystyle y=b^{log _{b}y}} in the left hand sides.

[1]

Formula Example
Product logb⁡(xy)=logb⁡x+logb⁡y{displaystyle log _{b}(xy)=log _{b}x+log _{b}y} log3⁡243=log3⁡(9⋅27)=log3⁡9+log3⁡27=2+3=5{displaystyle log _{3}243=log _{3}(9cdot 27)=log _{3}9+log _{3}27=2+3=5}
Quotient logbxy=logb⁡x−logb⁡y{displaystyle log _{b}!{frac {x}{y}}=log _{b}x-log _{b}y} log2⁡16=log2644=log2⁡64−log2⁡4=6−2=4{displaystyle log _{2}16=log _{2}!{frac {64}{4}}=log _{2}64-log _{2}4=6-2=4}
Power logb⁡(xp)=plogb⁡x{displaystyle log _{b}left(x^{p}right)=plog _{b}x} log2⁡64=log2⁡(26)=6log2⁡2=6{displaystyle log _{2}64=log _{2}left(2^{6}right)=6log _{2}2=6}
Root logb⁡xp=logb⁡xp{displaystyle log _{b}{sqrt[{p}]{x}}={frac {log _{b}x}{p}}} log10⁡1000=12log10⁡1000=32=1.5{displaystyle log _{10}{sqrt {1000}}={frac {1}{2}}log _{10}1000={frac {3}{2}}=1.5}

Change of base[

edit

]

The logarithm logbx can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:

logb⁡x=logk⁡xlogk⁡b.{displaystyle log _{b}x={frac {log _{k}x}{log _{k}b}}.,}
Derivation of the conversion factor between logarithms of arbitrary base

Starting from the defining identity

x=blogb⁡x{displaystyle x=b^{log _{b}x}}

we can apply logk to both sides of this equation, to get

logk⁡x=logk⁡(blogb⁡x)=logb⁡x⋅logk⁡b{displaystyle log _{k}x=log _{k}left(b^{log _{b}x}right)=log _{b}xcdot log _{k}b}.

Solving for logb⁡x{displaystyle log _{b}x} yields:

logb⁡x=logk⁡xlogk⁡b{displaystyle log _{b}x={frac {log _{k}x}{log _{k}b}}},

showing the conversion factor from given logk{displaystyle log _{k}}-values to their corresponding logb{displaystyle log _{b}}-values to be (logk⁡b)−1.{displaystyle (log _{k}b)^{-1}.}

Typical

scientific calculators

calculate the logarithms to bases 10 and

e

.

[7]

Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:

logb⁡x=log10⁡xlog10⁡b=loge⁡xloge⁡b.{displaystyle log _{b}x={frac {log _{10}x}{log _{10}b}}={frac {log _{e}x}{log _{e}b}}.,}

Given a number x and its logarithm y = logbx to an unknown base b, the base is given by:

b=x1y,{displaystyle b=x^{frac {1}{y}},}

which can be seen from taking the defining equation x=blogb⁡x=by{displaystyle x=b^{log _{b}x}=b^{y}} to the power of 1y.{displaystyle ;{tfrac {1}{y}}.}

Particular bases[

edit

]

Plots of logarithm for bases 0.5, 2, and e

Among all choices for the base, three are particularly common. These are b = 10, b =

e

(the

irrational

mathematical constant ≈ 2.71828), and b = 2 (the

binary logarithm

). In

mathematical analysis

, the logarithm base e is widespread because of analytical properties explained below. On the other hand, base-10 logarithms are easy to use for manual calculations in the

decimal

number system:

[8]

log10⁡(10x)=log10⁡10+log10⁡x=1+log10⁡x. {displaystyle log _{10}(10x)=log _{10}10+log _{10}x=1+log _{10}x. }

Thus, log10x is related to the number of

decimal digits

of a positive integer x: the number of digits is the smallest

integer

strictly bigger than log10x.

[9]

For example, log101430 is approximately 3.15. The next integer is 4, which is the number of digits of 1430. Both the natural logarithm and the logarithm to base two are used in

information theory

, corresponding to the use of

nats

or

bits

as the fundamental units of information, respectively.

[10]

Binary logarithms are also used in

computer science

, where the

binary system

is ubiquitous; in

music theory

, where a pitch ratio of two (the

octave

) is ubiquitous and the

cent

is the binary logarithm (scaled by 1200) of the ratio between two adjacent equally-tempered pitches in European

classical music

; and in

photography

to measure

exposure values

.

[11]

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 logbx, when the intended base can be determined from the context. The notation blog x also occurs.

[12]

The “ISO notation” column lists designations suggested by the

International Organization for Standardization

(

ISO 80000-2

).

[13]

Because the notation log x has been used for all three bases (or when the base is indeterminate or immaterial), the intended base must often be inferred based on context or discipline. In computer science log usually refers to log2, and in mathematics log usually refers to loge.

[14]

[1]

In other contexts log often means log10.

[15]

Base b Name for logbx ISO notation Other notations Used in
2

binary logarithm

lb x

[16]

ld x, log x, lg x,

[17]

log2x

computer science

,

information theory

,

bioinformatics

,

music theory

,

photography

e

natural logarithm

ln x

[nb 2]

log x
(in mathematics

[1]

[21]

and many

programming languages

[nb 3]

), logex

mathematics, physics, chemistry,

statistics

,

economics

, information theory, and engineering

10

common logarithm

lg x log x, log10x
(in engineering, biology, astronomy)
various

engineering

fields (see

decibel

and see below),
logarithm

tables

, handheld

calculators

,

spectroscopy

b logarithm to base b logbx mathematics

History[

edit

]

The history of logarithms in seventeenth-century Europe is the discovery of a new

function

that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by

John Napier

in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).

[22]

[23]

Prior to Napier’s invention, there had been other techniques of similar scopes, such as the

prosthaphaeresis

or the use of tables of progressions, extensively developed by

Jost Bürgi

around 1600.

[24]

[25]

Napier coined the term for logarithm in Middle Latin, “logarithmus,” derived from the Greek, literally meaning, “ratio-number,” from logos “proportion, ratio, word” + arithmos “number”.

The

common logarithm

of a number is the index of that power of ten which equals the number.

[26]

Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by

Archimedes

as the “order of a number”.

[27]

The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities.

[28]

Such methods are called

prosthaphaeresis

.

Invention of the

function

now known as the

natural logarithm

began as an attempt to perform a

quadrature

of a rectangular

hyperbola

by

Grégoire de Saint-Vincent

, a Belgian Jesuit residing in Prague. Archimedes had written

The Quadrature of the Parabola

in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a

geometric progression

in its

argument

and an

arithmetic progression

of values, prompted

A. A. de Sarasa

to make the connection of Saint-Vincent’s quadrature and the tradition of logarithms in

prosthaphaeresis

, leading to the term “hyperbolic logarithm”, a synonym for natural logarithm. Soon the new function was appreciated by

Christiaan Huygens

, and

James Gregory

. The notation Log y was adopted by

Leibniz

in 1675,

[29]

and the next year he connected it to the

integral

dyy.{displaystyle int {frac {dy}{y}}.}

Before Euler developed his modern conception of complex natural logarithms,

Roger Cotes

had a nearly equivalent result when he showed in 1714 that

[30]

log⁡(cos⁡θ+isin⁡θ)=iθ{displaystyle log(cos theta +isin theta )=itheta }

Logarithm tables, slide rules, and historical applications[

edit

]

The 1797

Encyclopædia Britannica

explanation of logarithms

By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially

astronomy

. They were critical to advances in

surveying

,

celestial navigation

, and other domains.

Pierre-Simon Laplace

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.”

[31]

As the function f(x) = bx is the inverse function of logbx, it has been called the antilogarithm.

[32]

Log tables[

edit

]

A key tool that enabled the practical use of logarithms was the

table of logarithms

.

[33]

The first such table was compiled by

Henry Briggs

in 1617, immediately after Napier’s invention but with the innovation of using 10 as the base. Briggs’ first table contained the

common logarithms

of all integers in the range 1–1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of log10x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an

integer part

and a

fractional part

, known as the characteristic and

mantissa

. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.

[34]

The characteristic of 10 · x is one plus the characteristic of x, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by

log10⁡3542=log10⁡(1000⋅3.542)=3+log10⁡3.542≈3+log10⁡3.54{displaystyle log _{10}3542=log _{10}(1000cdot 3.542)=3+log _{10}3.542approx 3+log _{10}3.54,}

Greater accuracy can be obtained by

interpolation

:

log10⁡3542≈3+log10⁡3.54+0.2(log10⁡3.55−log10⁡3.54){displaystyle log _{10}3542approx 3+log _{10}3.54+0.2(log _{10}3.55-log _{10}3.54),}

The value of 10x can be determined by reverse look up in the same table, since the logarithm is a

monotonic function

.

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Computations[

edit

]

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, via the same table:

cd=10 log10⁡c10 log10⁡d=10 log10⁡c+ log10⁡d{displaystyle cd=10^{ log _{10}c},10^{ log _{10}d}=10^{ log _{10}c+ log _{10}d},}

and

cd=cd−1=10 log10⁡c− log10⁡d.{displaystyle {frac {c}{d}}=cd^{-1}=10^{ log _{10}c- log _{10}d}.,}

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

cd=(10 log10⁡c)d=10d log10⁡c{displaystyle c^{d}=left(10^{ log _{10}c}right)^{d}=10^{d log _{10}c},}

and

cd=c1d=101d log10⁡c.{displaystyle {sqrt[{d}]{c}}=c^{frac {1}{d}}=10^{{frac {1}{d}} log _{10}c}.,}

Trigonometric calculations were facilitated by tables that contained the common logarithms of

trigonometric functions

.

Slide rules[

edit

]

Another critical application was the

slide rule

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

A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6.

Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x.

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 precision, much faster computation than techniques based on tables.

[35]

Analytic properties[

edit

]

A deeper study of logarithms requires the concept of a

function

. A function is a rule that, given one number, produces another number.

[36]

An example is the function producing the x-th power of b from any real number x, where the base b is a fixed number. This function is written: f(x)=bx.{displaystyle f(x)=b^{x}.,}

Logarithmic function[

edit

]

To justify the definition of logarithms, it is necessary to show that the equation

bx=y{displaystyle b^{x}=y,}

has 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 theorem

from elementary

calculus

.

[37]

This theorem states that a

continuous function

that 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) = bx. Because f takes arbitrarily large and arbitrarily small positive values, any number y > 0 lies between f(x0) and f(x1) for suitable x0 and x1. 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 increasing

(for b > 1), or strictly decreasing (for 0 < b < 1).

[38]

The unique solution x is the logarithm of y to base b, logby. The function that assigns to y its logarithm is called logarithm function or logarithmic function (or just logarithm).

The function logbx is essentially characterized by the product formula

logb⁡(xy)=logb⁡x+logb⁡y.{displaystyle log _{b}(xy)=log _{b}x+log _{b}y.}

More precisely, the logarithm to any base b > 1 is the only

increasing function

f from the positive reals to the reals satisfying f(b) = 1 and

[39]

f(xy)=f(x)+f(y).{displaystyle f(xy)=f(x)+f(y).}

Inverse function[

edit

]

The graphs of two functions.

The graph of the logarithm function logb(x) (blue) is obtained by

reflecting

the graph of the function bx (red) at the diagonal line (x = y).

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

logb⁡(bx)=xlogb⁡b=x.{displaystyle log _{b}left(b^{x}right)=xlog _{b}b=x.}

In prose, taking the x-th power of b and then the base-b logarithm gives back x. Conversely, given a positive number y, the formula

blogb⁡y=y{displaystyle b^{log _{b}y}=y}

says 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 f(x) = bx.

[40]

Inverse functions are closely related to the original functions. Their

graphs

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 = bt) on the graph of f yields a point (u, t = logbu) on the graph of the logarithm and vice versa. As a consequence, logb(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, logb(x) is an

increasing function

. For b < 1, logb(x) tends to minus infinity instead. When x approaches zero, logbx goes to minus infinity for b > 1 (plus infinity for b < 1, respectively).

Derivative and antiderivative[

edit

]

A graph of the logarithm function and a line touching it in one point.

The graph of the

natural logarithm

(green) and its tangent at x = 1.5 (black)

Analytic properties of functions pass to their inverses.

[37]

Thus, as f(x) = bx is a continuous and

differentiable function

, so is logby. Roughly, a continuous function is differentiable if its graph has no sharp “corners”. Moreover, as the

derivative

of f(x) evaluates to ln(b)bx by the properties of the

exponential function

, the

chain rule

implies that the derivative of logbx is given by

[38]

[41]

ddxlogb⁡x=1xln⁡b.{displaystyle {frac {d}{dx}}log _{b}x={frac {1}{xln b}}.}

That is, the

slope

of the

tangent

touching the graph of the base-b logarithm at the point (x, logb(x)) equals 1/(x ln(b)).

The derivative of ln x is 1/x; this implies that ln x is the unique

antiderivative

of 1/x that has the value 0 for x =1. It is this very simple formula that motivated to qualify as “natural” the natural logarithm; this is also one of the main reasons of the importance of the constant

e

.

The derivative with a generalised functional argument f(x) is

ddxln⁡f(x)=f′(x)f(x).{displaystyle {frac {d}{dx}}ln f(x)={frac {f'(x)}{f(x)}}.}

The quotient at the right hand side is called the

logarithmic derivative

of f. Computing f’(x) by means of the derivative of ln(f(x)) is known as

logarithmic differentiation

.

[42]

The antiderivative of the

natural logarithm

ln(x) is:

[43]

ln⁡(x)dx=xln⁡(x)−x+C.{displaystyle int ln(x),dx=xln(x)-x+C.}

Related formulas

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

[44]

Integral representation of the natural logarithm[

edit

]

A hyperbola with part of the area underneath shaded in grey.

The

natural logarithm

of t is the shaded area underneath the graph of the function f(x) = 1/x (reciprocal of x).

The

natural logarithm

of t can be defined as the

definite integral

:

ln⁡t=∫1t1xdx.{displaystyle ln t=int _{1}^{t}{frac {1}{x}},dx.}

This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, ln(t) equals the area between the x-axis and the graph of the function 1/x, ranging from x = 1 to x = t. This is a consequence of the

fundamental theorem of calculus

and the fact that the derivative of ln(x) is 1/x. Product and power logarithm formulas can be derived from this definition.

[45]

For example, the product formula ln(tu) = ln(t) + ln(u) is deduced as:

ln⁡(tu)=∫1tu1xdx =(1)∫1t1xdx+∫ttu1xdx =(2)ln⁡(t)+∫1u1wdw=ln⁡(t)+ln⁡(u).{displaystyle ln(tu)=int _{1}^{tu}{frac {1}{x}},dx {stackrel {(1)}{=}}int _{1}^{t}{frac {1}{x}},dx+int _{t}^{tu}{frac {1}{x}},dx {stackrel {(2)}{=}}ln(t)+int _{1}^{u}{frac {1}{w}},dw=ln(t)+ln(u).}

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 hyperbola depicted twice. The area underneath is split into different parts.

A visual proof of the product formula of the natural logarithm

The power formula ln(tr) = r ln(t) may be derived in a similar way:

ln⁡(tr)=∫1tr1xdx=∫1t1wr(rwr−1dw)=r∫1t1wdw=rln⁡(t).{displaystyle ln(t^{r})=int _{1}^{t^{r}}{frac {1}{x}}dx=int _{1}^{t}{frac {1}{w^{r}}}left(rw^{r-1},dwright)=rint _{1}^{t}{frac {1}{w}},dw=rln(t).}

The second equality uses a change of variables (

integration by substitution

), w = x1/r.

The sum over the reciprocals of natural numbers,

1+12+13+⋯+1n=∑k=1n1k,{displaystyle 1+{frac {1}{2}}+{frac {1}{3}}+cdots +{frac {1}{n}}=sum _{k=1}^{n}{frac {1}{k}},}

is called the

harmonic series

. It is closely tied to the

natural logarithm

: as n tends to

infinity

, the difference,

k=1n1k−ln⁡(n),{displaystyle sum _{k=1}^{n}{frac {1}{k}}-ln(n),}

converges

(i.e., gets arbitrarily close) to a number known as the

Euler–Mascheroni constant

γ = 0.5772…. This relation aids in analyzing the performance of algorithms such as

quicksort

.

[46]

Transcendence of the logarithm[

edit

]

Real numbers

that are not

algebraic

are called

transcendental

;

[47]

for example,

π

and

e

are such numbers, but 2−3{displaystyle {sqrt {2-{sqrt {3}}}}} is not.

Almost all

real numbers are transcendental. The logarithm is an example of a

transcendental function

. The

Gelfond–Schneider theorem

asserts that logarithms usually take transcendental, i.e., “difficult” values.

[48]

Calculation[

edit

]

The logarithm keys (LOG for base-10 and LN for base-e) on a

TI-83 Plus

graphing calculator.

Logarithms are easy to compute in some cases, such as log10(1000) = 3. In general, logarithms can be calculated using

power series

or the

arithmetic–geometric mean

, or be retrieved from a precalculated

logarithm table

that provides a fixed precision.

[49]

[50]

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.

[51]

Using look-up tables,

CORDIC

-like methods can be used to compute logarithms by using only the operations of addition and

bit shifts

.

[52]

[53]

Moreover, the

binary logarithm algorithm

calculates lb(x)

recursively

, based on repeated squarings of x, taking advantage of the relation

log2⁡(x2)=2log2⁡|x|.{displaystyle log _{2}left(x^{2}right)=2log _{2}|x|.}

Xem thêm: Các dạng toán về đạo hàm của hàm số, cách tính và bài tập áp dụng-toán lớp 11

Power series[

edit

]

Taylor series

An animation showing increasingly good approximations of the logarithm graph.

The Taylor series of ln(z) centered at z = 1. The animation shows the first 10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.

For any real number z that satisfies 0 < z ≤ 2, the following formula holds:

[nb 4]

[54]

ln⁡(z)=(z−1)11−(z−1)22+(z−1)33−(z−1)44+⋯=∑k=1∞(−1)k+1(z−1)kk{displaystyle {begin{aligned}ln(z)&={frac {(z-1)^{1}}{1}}-{frac {(z-1)^{2}}{2}}+{frac {(z-1)^{3}}{3}}-{frac {(z-1)^{4}}{4}}+cdots \&=sum _{k=1}^{infty }(-1)^{k+1}{frac {(z-1)^{k}}{k}}end{aligned}}}

This is a shorthand for saying that ln(z) can be approximated to a more and more accurate value by the following expressions:

(z−1)(z−1)−(z−1)22(z−1)−(z−1)22+(z−1)33⋮{displaystyle {begin{array}{lllll}(z-1)&&\(z-1)&-&{frac {(z-1)^{2}}{2}}&\(z-1)&-&{frac {(z-1)^{2}}{2}}&+&{frac {(z-1)^{3}}{3}}\vdots &end{array}}}

For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465. This

series

approximates ln(z) with arbitrary precision, provided the number of summands is large enough. In elementary calculus, ln(z) is therefore the

limit

of this series. It is the

Taylor series

of the

natural logarithm

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

ln⁡(1+z)=z−z22+z33⋯z.{displaystyle ln(1+z)=z-{frac {z^{2}}{2}}+{frac {z^{3}}{3}}cdots approx z.}

For example, with z = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953.

More efficient series

Another series is based on the

area hyperbolic tangent

function:

ln⁡(z)=2⋅artanhz−1z+1=2(z−1z+1+13(z−1z+1)3+15(z−1z+1)5+⋯),{displaystyle ln(z)=2cdot operatorname {artanh} ,{frac {z-1}{z+1}}=2left({frac {z-1}{z+1}}+{frac {1}{3}}{left({frac {z-1}{z+1}}right)}^{3}+{frac {1}{5}}{left({frac {z-1}{z+1}}right)}^{5}+cdots right),}

for any real number z > 0.

[nb 5]

[54]

Using

sigma notation

, this is also written as

ln⁡(z)=2∑k=0∞12k+1(z−1z+1)2k+1.{displaystyle ln(z)=2sum _{k=0}^{infty }{frac {1}{2k+1}}left({frac {z-1}{z+1}}right)^{2k+1}.}

This 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 3×10−6. The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting

A=zexp⁡(y),{displaystyle A={frac {z}{exp(y)}},,}

the logarithm of z is:

ln⁡(z)=y+ln⁡(A).{displaystyle ln(z)=y+ln(A).,}

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 series

, 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 z = a · 10b, so that ln(z) = ln(a) + b · ln(10).

A closely related method can be used to compute the logarithm of integers. Putting z=n+1n{displaystyle textstyle z={frac {n+1}{n}}} in the above series, it follows that:

ln⁡(n+1)=ln⁡(n)+2∑k=0∞12k+1(12n+1)2k+1.{displaystyle ln(n+1)=ln(n)+2sum _{k=0}^{infty }{frac {1}{2k+1}}left({frac {1}{2n+1}}right)^{2k+1}.}

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

rate of convergence

of 12n+1{displaystyle {frac {1}{2n+1}}}.

Arithmetic–geometric mean approximation[

edit

]

The

arithmetic–geometric mean

yields high precision approximations of the

natural logarithm

. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work ln(x) is approximated to a precision of 2p (or p precise bits) by the following formula (due to

Carl Friedrich Gauss

):

[55]

[56]

ln⁡(x)≈π2M(1,22−m/x)−mln⁡(2).{displaystyle ln(x)approx {frac {pi }{2M(1,2^{2-m}/x)}}-mln(2).}

Here M(x,y) denotes the

arithmetic–geometric mean

of x and y. It is obtained by repeatedly calculating the average (x+y)/2{displaystyle (x+y)/2} (

arithmetic mean

) and xy{displaystyle {sqrt {xy}}} (

geometric mean

) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of M(x,y). m is chosen such that

x2m>2p/2.{displaystyle x,2^{m}>2^{p/2}.,}

to ensure the required precision. A larger m makes the M(x,y) calculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision. The constants pi and ln(2) can be calculated with quickly converging series.

Feynman’s algorithm[

edit

]

While at

Los Alamos National Laboratory

working on the

Manhattan Project

,

Richard Feynman

developed a bit-processing algorithm that is similar to long division and was later used in the

Connection Machine

. The algorithm uses the fact that every real number 1<x<2{displaystyle 1<x<2} is representable as a product of distinct factors of the form 1+2−k{displaystyle 1+2^{-k}}. The algorithm sequentially builds that product P{displaystyle P}: if P⋅(1+2−k)<x{displaystyle Pcdot (1+2^{-k})<x}, then it changes P{displaystyle P} to P⋅(1+2−k){displaystyle Pcdot (1+2^{-k})}. It then increases k{displaystyle k} by one regardless. The algorithm stops when k{displaystyle k} is large enough to give the desired accuracy. Because log⁡(x){displaystyle log(x)} is the sum of the terms of the form log⁡(1+2−k){displaystyle log(1+2^{-k})} corresponding to those k{displaystyle k} for which the factor 1+2−k{displaystyle 1+2^{-k}} was included in the product P{displaystyle P}, log⁡(x){displaystyle log(x)} may be computed by simple addition, using a table of log⁡(1+2−k){displaystyle log(1+2^{-k})} for all k{displaystyle k}. Any base may be used for the logarithm table.

[57]

Applications[

edit

]

A photograph of a nautilus' shell.

A

nautilus

displaying a logarithmic spiral

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

scale invariance

. 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

.

[58]

Benford’s law

on the distribution of leading digits can also be explained by scale invariance.

[59]

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.

[60]

The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms.

Logarithmic scales

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 equation

, the

Fenske equation

, or the

Nernst equation

.

Logarithmic scale[

edit

]

A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale.

A logarithmic chart depicting the value of one

Goldmark

in

Papiermarks

during the

German hyperinflation in the 1920s

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

decibel

is a

unit of measurement

associated with

logarithmic-scale

quantities

. It is based on the common logarithm of

ratios

—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,

[61]

to describe power levels of sounds in

acoustics

,

[62]

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.

[63]

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.

[64]

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 magnitude scale

. For example, a 5.0 earthquake releases 32 times (101.5) and a 6.0 releases 1000 times (103) the energy of a 4.0.

[65]

Another logarithmic scale is

apparent magnitude

. It measures the brightness of stars logarithmically.

[66]

Yet another example is

pH

in

chemistry

; pH is the negative of the common logarithm of the

activity

of

hydronium

ions (the form

hydrogen

ions

H+
take in water).

[67]

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 104 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 functions

of the form f(x) = a · bx appear as straight lines with

slope

equal to the logarithm of b.

Log-log

graphs scale both axes logarithmically, which causes functions of the form f(x) = a · xk to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing

power laws

.

[68]

Psychology[

edit

]

Logarithms occur in several laws describing

human perception

:

[69]

[70]

Hick’s law

proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.

[71]

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.

[72]

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.

[73]

(This “law”, however, is less realistic than more recent models, such as

Stevens’s power law

.

[74]

)

Psychological studies found that individuals with little mathematics education 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 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.

[75]

[76]

Probability theory and statistics[

edit

]

Three asymmetric PDF curves

Three

probability density functions

(PDF) of random variables with log-normal distributions. The location parameter μ, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.

A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion.

Distribution of first digits (in %, red bars) in the

population of the 237 countries

of the world. Black dots indicate the distribution predicted by Benford’s law.

Logarithms arise in

probability theory

: 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

.

[77]

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.

[78]

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.

[79]

Logarithms are used for

maximum-likelihood estimation

of parametric

statistical models

. For such a model, the

likelihood function

depends on at least one

parameter

that must 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.

[80]

Benford’s law

describes the occurrence of digits in many

data sets

, 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 log10(d + 1) − log10(d), regardless of the unit of measurement.

[81]

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.

[82]

Computational complexity[

edit

]

Analysis of algorithms

is a branch of

computer science

that studies the

performance

of

algorithms

(computer programs solving a certain problem).

[83]

Logarithms are valuable for describing algorithms that

divide a problem

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

[84]

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, log2(N) comparisons, where N is the list’s length.

[85]

Similarly, the

merge sort

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

N · log(N).

[86]

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

.

[87]

A function f(x) is said to

grow logarithmically

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.

[88]

) For example, any

natural number

N can be represented in

binary form

in no more than log2(N) + 1

bits

. In other words, the amount of

memory

needed to store N grows logarithmically with N.

Xem thêm: Nhận biết bún, bánh phở, bánh canh tươi… bị ô nhiễm chất phát quang Tinopal

Entropy and chaos[

edit

]

An oval shape with the trajectories of two particles.

Billiards

on an oval

billiard table

. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of

reflections

at the boundary.

Entropy

is broadly a measure of the disorder of some system. In

statistical thermodynamics

, the entropy S of some physical system is defined as

S=−k∑ipiln⁡(pi).{displaystyle S=-ksum _{i}p_{i}ln(p_{i}).,}

The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, pi 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 log2(N) bits.

[89]

Lyapunov exponents

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 measurement errors of the initial state predictably lead to largely different final states.

[90]

At least one Lyapunov exponent of a deterministically chaotic system is positive.

Fractals[

edit

]

Parts of a triangle are removed in an iterated way.

The Sierpinski triangle (at the right) is constructed by repeatedly replacing

equilateral triangles

by three smaller ones.

Logarithms occur in definitions of the

dimension

of

fractals

.

[91]

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 makes the

Hausdorff dimension

of this structure ln(3)/ln(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[

edit

]

Four different octaves shown on a linear scale.

Four different octaves shown on a logarithmic scale.

Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them).

Logarithms are related to musical tones and

intervals

. 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 A

has a frequency of 440

Hz

and

B-flat

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

semitone

, as is the one between B-flat and

B

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

466440≈493466≈1.059≈212.{displaystyle {frac {466}{440}}approx {frac {493}{466}}approx 1.059approx {sqrt[{12}]{2}}.}

Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the base-21/12 logarithm of the

frequency

ratio, while the base-21/1200 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.

[92]

Interval
(the two tones are played at the same time)

1/12 tone

About this sound

play

 

Semitone

About this sound

play

Just major third

About this sound

play

Major third

About this sound

play

Tritone

About this sound

play

Octave

About this sound

play

Frequency ratio r 2172≈1.0097{displaystyle 2^{frac {1}{72}}approx 1.0097} 2112≈1.0595{displaystyle 2^{frac {1}{12}}approx 1.0595} 54=1.25{displaystyle {tfrac {5}{4}}=1.25} 2412=23≈1.2599{displaystyle {begin{aligned}2^{frac {4}{12}}&={sqrt[{3}]{2}}\&approx 1.2599end{aligned}}} 2612=2≈1.4142{displaystyle {begin{aligned}2^{frac {6}{12}}&={sqrt {2}}\&approx 1.4142end{aligned}}} 21212=2{displaystyle 2^{frac {12}{12}}=2}
Corresponding number of semitones
log212⁡(r)=12log2⁡(r){displaystyle log _{sqrt[{12}]{2}}(r)=12log _{2}(r)}
16{displaystyle {tfrac {1}{6}},} 1{displaystyle 1,} 3.8631{displaystyle approx 3.8631,} 4{displaystyle 4,} 6{displaystyle 6,} 12{displaystyle 12,}
Corresponding number of cents
log21200⁡(r)=1200log2⁡(r){displaystyle log _{sqrt[{1200}]{2}}(r)=1200log _{2}(r)}
1623{displaystyle 16{tfrac {2}{3}},} 100{displaystyle 100,} 386.31{displaystyle approx 386.31,} 400{displaystyle 400,} 600{displaystyle 600,} 1200{displaystyle 1200,}

Number theory[

edit

]

Natural logarithms

are closely linked to

counting prime numbers

(2, 3, 5, 7, 11, …), an important topic in

number theory

. For any

integer

x, the quantity of

prime numbers

less than or equal to x is denoted

π(x)

. The

prime number theorem

asserts that π(x) is approximately given by

xln⁡(x),{displaystyle {frac {x}{ln(x)}},}

in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity.

[93]

As a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely

proportional

to the number of decimal digits of x. A far better estimate of π(x) is given by the

offset logarithmic integral

function Li(x), defined by

Li(x)=∫2x1ln⁡(t)dt.{displaystyle mathrm {Li} (x)=int _{2}^{x}{frac {1}{ln(t)}},dt.}

The

Riemann hypothesis

, one of the oldest open mathematical

conjectures

, can be stated in terms of comparing π(x) and Li(x).

[94]

The

Erdős–Kac theorem

describing the number of distinct

prime factors

also involves the

natural logarithm

.

The logarithm of n

factorial

, n! = 1 · 2 · … · n, is given by

ln⁡(n!)=ln⁡(1)+ln⁡(2)+⋯+ln⁡(n).{displaystyle ln(n!)=ln(1)+ln(2)+cdots +ln(n).,}

This can be used to obtain

Stirling’s formula

, an approximation of n! for large n.

[95]

Generalizations[

edit

]

Complex logarithm[

edit

]

An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x axis.

Polar form of z = x + iy. Both φ and φ’ are arguments of z.

All the

complex numbers

a that solve the equation

ea=z{displaystyle e^{a}=z}

are called complex logarithms of z, when z is (considered as) a complex number. A complex number is commonly represented as z = x + iy, where x and y are real numbers and i is an

imaginary unit

, the square of which is −1. 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 z by its

absolute value

, that is, the (positive, real) distance r to the

origin

, and an angle between the real (x) axis Re and the line passing through both the origin and z. This angle is called the

argument

of z.

The absolute value r of z is given by

r=x2+y2.{displaystyle textstyle r={sqrt {x^{2}+y^{2}}}.}

Using the geometrical interpretation of sin{displaystyle sin } and cos{displaystyle cos } and their periodicity in ,{displaystyle 2pi ,} any complex number z may be denoted as

z=x+iy=r(cos⁡φ+isin⁡φ)=r(cos⁡+2kπ)+isin⁡+2kπ)),{displaystyle z=x+iy=r(cos varphi +isin varphi )=r(cos(varphi +2kpi )+isin(varphi +2kpi )),}

for any integer number k. Evidently the argument of z is not uniquely specified: both φ and φ‘ = φ + 2kπ are valid arguments of z for all integers k, because adding 2kπ

radian

or k⋅360°

[nb 6]

to φ corresponds to “winding” around the origin counter-clock-wise by k

turns

. The resulting complex number is always z, as illustrated at the right for k = 1. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g., ππ{displaystyle -pi <varphi leq pi }

[96]

or 0≤φ<2π.{displaystyle 0leq varphi <2pi .}

[97]

These regions, where the argument of z is uniquely determined are called

branches

of the argument function.

A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.

The principal branch (-π, π) of the complex logarithm, Log(z). The black point at z = 1 corresponds to absolute value zero and brighter, more

saturated

colors refer to bigger absolute values. The

hue

of the color encodes the argument of Log(z).

Euler’s formula

connects the

trigonometric functions

sine

and

cosine

to the

complex exponential

:

eiφ=cos⁡φ+isin⁡φ.{displaystyle e^{ivarphi }=cos varphi +isin varphi .}

Using this formula, and again the periodicity, the following identities hold:

[98]

z=r(cos⁡φ+isin⁡φ)=r(cos⁡+2kπ)+isin⁡+2kπ))=rei(φ+2kπ)=eln⁡(r)ei(φ+2kπ)=eln⁡(r)+i(φ+2kπ)=eak,{displaystyle {begin{array}{lll}z&=&rleft(cos varphi +isin varphi right)\&=&rleft(cos(varphi +2kpi )+isin(varphi +2kpi )right)\&=&re^{i(varphi +2kpi )}\&=&e^{ln(r)}e^{i(varphi +2kpi )}\&=&e^{ln(r)+i(varphi +2kpi )}=e^{a_{k}},end{array}}}

where ln(r) is the unique real natural logarithm, ak denote the complex logarithms of z, and k is an arbitrary integer. Therefore, the complex logarithms of z, which are all those complex values ak for which the ak-th power of e equals z, are the infinitely many values

ak=ln⁡(r)+i(φ+2kπ),{displaystyle a_{k}=ln(r)+i(varphi +2kpi ),quad } for arbitrary integers k.

Taking k such that φ+2kπ{displaystyle varphi +2kpi } is within the defined interval for the principal arguments, then ak is called the principal value of the logarithm, denoted Log(z), again with a capital L. 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.

[99]

The illustration at the right depicts Log(z), confining the arguments of z to the interval (-π, π]. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e., not changing to the corresponding k-value of the continuously neighboring branch. Such a locus is called a

branch cut

. Dropping the range restrictions on the argument makes the relations “argument of z“, and consequently the “logarithm of z“,

multi-valued functions

.

Inverses of other exponential functions[

edit

]

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

logarithm of a matrix

is the (multi-valued) inverse function of the

matrix exponential

.

[100]

Another example is the

p-adic logarithm

, the inverse function of the

p-adic exponential

. Both are defined via Taylor series analogous to the real case.

[101]

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.

[102]

In the context of

finite groups

exponentiation is given by repeatedly multiplying one group element b with itself. The

discrete logarithm

is the integer n solving the equation

bn=x,{displaystyle b^{n}=x,,}

where 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

cryptographic

keys over unsecured information channels.

[103]

Zech’s logarithm

is related to the discrete logarithm in the multiplicative group of non-zero elements of a

finite field

.

[104]

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 logarithm

in computer science), the

Lambert W function

, and the

logit

. They are the inverse functions of the

double exponential function

,

tetration

, of f(w) = wew,

[105]

and of the

logistic function

, respectively.

[106]

Related concepts[

edit

]

From the perspective of

group theory

, the identity log(cd) = log(c) + log(d) expresses a

group isomorphism

between positive

reals

under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.

[107]

By means of that isomorphism, the

Haar measure

(

Lebesgue measure

) dx on the reals corresponds to the Haar measure dx/x on the positive reals.

[108]

The non-negative reals not only have a multiplication, but also have addition, and form a

semiring

, called the

probability semiring

; this is in fact a

semifield

. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (

LogSumExp

), giving an

isomorphism

of semirings between the probability semiring and the

log semiring

.

Logarithmic one-forms

df/f appear in

complex analysis

and

algebraic geometry

as

differential forms

with logarithmic

poles

.

[109]

The

polylogarithm

is the function defined by

Lis⁡(z)=∑k=1∞zkks.{displaystyle operatorname {Li} _{s}(z)=sum _{k=1}^{infty }{z^{k} over k^{s}}.}

It is related to the

natural logarithm

by Li1(z) = −ln(1 − z). Moreover, Lis(1) equals the

Riemann zeta function

ζ(s).

[110]

See also[

edit

]

  • Cologarithm

  • Decimal exponent

    (dex)

  • Exponential function

  • Index of logarithm articles

  • Logarithmic notation

Notes[

edit

]

  1. ^

    The restrictions on x and b are explained in the section

    “Analytic properties”

    .

  2. ^

    Some mathematicians disapprove of this notation. In his 1985 autobiography,

    Paul Halmos

    criticized what he considered the “childish ln notation,” which he said no mathematician had ever used.

    [18]

    The notation was invented by

    Irving Stringham

    , a mathematician.

    [19]

    [20]

  3. ^

    For example

    C

    ,

    Java

    ,

    Haskell

    , and

    BASIC

    .

  4. ^

    The same series holds for the principal value of the complex logarithm for complex numbers z satisfying |z − 1| < 1.

  5. ^

    The same series holds for the principal value of the complex logarithm for complex numbers z with positive real part.

  6. ^

    See

    radian

    for the conversion between 2

    π

    and 360

    degree

    .

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edit

]

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External links[

edit

]

  • Media related to

    Logarithm

    at Wikimedia Commons

  • The dictionary definition of

    logarithm

    at Wiktionary

  • Weisstein, Eric W.

    ,

    “Logarithm”

    ,

    MathWorld

  • Khan Academy: Logarithms, free online micro lectures

  • “Logarithmic function”

    ,

    Encyclopedia of Mathematics

    ,

    EMS Press

    , 2001 [1994]

  • Colin Byfleet,

    Educational video on logarithms

    , retrieved 12 October 2010

  • Edward Wright,

    Translation of Napier’s work on logarithms

    , archived from the original on 3 December 2002, retrieved 12 October 2010CS1 maint: unfit URL (

    link

    )

  • Glaisher, James Whitbread Lee (1911),

    “Logarithm” 

    , in Chisholm, Hugh (ed.),

    Encyclopædia Britannica

    , 16 (11th ed.), Cambridge University Press, pp. 868–77

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