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Arithmetic operations 


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 = 10^{3}, the “logarithm base 10” of 1000 is 3, or log_{10}(1000) = 3. The logarithm of x to base b is denoted as log_{b}(x), or without parentheses, log_{b} x, 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 log_{b}(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, log_{2} 64 = 6, as 2^{6} = 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 highaccuracy computations more easily. Using
logarithm tables
, tedious multidigit multiplication steps can be replaced by table lookups 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)=logbx+logby,{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 presentday 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 wideranging 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
publickey cryptography
.
Motivation and definition[
edit
]
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 nextsimplest 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=2×2×2=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 yth 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:
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 nth power, where n is a
natural number
, is done by multiplying n factors equal to b. The nth power of b is written b^{n}, so that
 bn=b×b×⋯×b⏟n factors{displaystyle b^{n}=underbrace {btimes btimes cdots times b} _{n{text{ factors}}}}
Exponentiation may be extended to b^{y}, 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 qth 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 yth 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 b^{m + n} = b^{m} · b^{n} 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 “log_{b} x” (pronounced as “the logarithm of x to base b” or “the baseb logarithm of x” or (most commonly) “the log, base b, of 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?”.
Examples[
edit
]
 log_{2} 16 = 4 , since 2^{4} = 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}}.}
 log_{10} 150 is approximately 2.176, which lies between 2 and 3, just as 150 lies between 10^{2} = 100 and 10^{3} = 1000.
 For any base b, log_{b} b = 1 and log_{b} 1 = 0, since b^{1} = b and b^{0} = 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 pth power of a number is p times the logarithm of the number itself; the logarithm of a pth 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=blogbx{displaystyle x=b^{log _{b}x}} or y=blogby{displaystyle y=b^{log _{b}y}} in the left hand sides.^{}
[1]
Formula  Example  

Product  logb(xy)=logbx+logby{displaystyle log _{b}(xy)=log _{b}x+log _{b}y}  log3243=log3(9⋅27)=log39+log327=2+3=5{displaystyle log _{3}243=log _{3}(9cdot 27)=log _{3}9+log _{3}27=2+3=5} 
Quotient  logbxy=logbx−logby{displaystyle log _{b}!{frac {x}{y}}=log _{b}xlog _{b}y}  log216=log2644=log264−log24=6−2=4{displaystyle log _{2}16=log _{2}!{frac {64}{4}}=log _{2}64log _{2}4=62=4} 
Power  logb(xp)=plogbx{displaystyle log _{b}left(x^{p}right)=plog _{b}x}  log264=log2(26)=6log22=6{displaystyle log _{2}64=log _{2}left(2^{6}right)=6log _{2}2=6} 
Root  logbxp=logbxp{displaystyle log _{b}{sqrt[{p}]{x}}={frac {log _{b}x}{p}}}  log101000=12log101000=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 log_{b}x can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:
 logbx=logkxlogkb.{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
we can apply log_{k} to both sides of this equation, to get
Solving for logbx{displaystyle log _{b}x} yields:
showing the conversion factor from given logk{displaystyle log _{k}}values to their corresponding logb{displaystyle log _{b}}values to be (logkb)−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:
 logbx=log10xlog10b=logexlogeb.{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 = log_{b} x 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=blogbx=by{displaystyle x=b^{log _{b}x}=b^{y}} to the power of 1y.{displaystyle ;{tfrac {1}{y}}.}
Particular bases[
edit
]
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, base10 logarithms are easy to use for manual calculations in the
decimal
number system:^{}
[8]
 log10(10x)=log1010+log10x=1+log10x. {displaystyle log _{10}(10x)=log _{10}10+log _{10}x=1+log _{10}x. }
Thus, log_{10} x is related to the number of
decimal digits
of a positive integer x: the number of digits is the smallest
integer
strictly bigger than log_{10} x.^{}
[9]
For example, log_{10}1430 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 equallytempered 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 log_{b} x, when the intended base can be determined from the context. The notation ^{b}log x also occurs.^{}
[12]
The “ISO notation” column lists designations suggested by the
International Organization for Standardization
(
ISO 800002
).^{}
[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 log_{2}, and in mathematics log usually refers to log_{e}.^{}
[14]
^{}
[1]
In other contexts log often means log_{10}.^{}
[15]
Base b  Name for log_{b} x  ISO notation  Other notations  Used in 

2 
binary logarithm 
lb x^{ [16] }  ld x, log x, lg x,^{ [17] log2 x } 
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] ), loge x } 
mathematics, physics, chemistry,
statistics , economics , information theory, and engineering 
10 
common logarithm 
lg x  log x, log_{10} x (in engineering, biology, astronomy) 
various
engineering fields (see decibel and see below), tables , handheld calculators , spectroscopy 
b  logarithm to base b  log_{b}x  mathematics 
History[
edit
]
The history of logarithms in seventeenthcentury 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, “rationumber,” 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 SaintVincent
, 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 SaintVincent 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 SaintVincent’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(cosθ+isinθ)=iθ{displaystyle log(cos theta +isin theta )=itheta }
Logarithm tables, slide rules, and historical applications[
edit
]
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.
PierreSimon 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) = b^{x} is the inverse function of log_{b} x, 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 log_{10} x for any number x in a certain range, at a certain precision. Base10 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 threedigit log table, the logarithm of 3542 is approximated by
 log103542=log10(1000⋅3.542)=3+log103.542≈3+log103.54{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
:
 log103542≈3+log103.54+0.2(log103.55−log103.54){displaystyle log _{10}3542approx 3+log _{10}3.54+0.2(log _{10}3.55log _{10}3.54),}
The value of 10^{x} 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 log10c10 log10d=10 log10c+ log10d{displaystyle cd=10^{ log _{10}c},10^{ log _{10}d}=10^{ log _{10}c+ log _{10}d},}
and
 cd=cd−1=10 log10c− log10d.{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 lookups by
 cd=(10 log10c)d=10d log10c{displaystyle c^{d}=left(10^{ log _{10}c}right)^{d}=10^{d log _{10}c},}
and
 cd=c1d=101d log10c.{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 nonsliding 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:
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 xth 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) = b^{ x}. Because f takes arbitrarily large and arbitrarily small positive values, any number y > 0 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 increasing
(for b > 1), or strictly decreasing (for 0 < b < 1).^{}
[38]
The unique solution x is the logarithm of y to base b, log_{b} y. The function that assigns to y its logarithm is called logarithm function or logarithmic function (or just logarithm).
The function log_{b} x is essentially characterized by the product formula
 logb(xy)=logbx+logby.{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 formula for the logarithm of a power says in particular that for any number x,
 logb(bx)=xlogbb=x.{displaystyle log _{b}left(b^{x}right)=xlog _{b}b=x.}
In prose, taking the xth power of b and then the baseb logarithm gives back x. Conversely, given a positive number y, the formula
 blogby=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) = b^{x}.^{}
[40]
Inverse functions are closely related to the original functions. Their
graphs
correspond to each other upon exchanging the x– and the ycoordinates (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 b < 1, log_{b}(x) tends to minus infinity instead. When x approaches zero, log_{b}x goes to minus infinity for b > 1 (plus infinity for b < 1, respectively).
Derivative and antiderivative[
edit
]
Analytic properties of functions pass to their inverses.^{}
[37]
Thus, as f(x) = b^{x} is a continuous and
differentiable function
, so is log_{b} y. Roughly, a continuous function is differentiable if its graph has no sharp “corners”. Moreover, as the
derivative
of f(x) evaluates to ln(b)b^{x} by the properties of the
exponential function
, the
chain rule
implies that the derivative of log_{b} x is given by^{}
[38]
^{}
[41]
 ddxlogbx=1xlnb.{displaystyle {frac {d}{dx}}log _{b}x={frac {1}{xln b}}.}
That is, the
slope
of the
tangent
touching the graph of the baseb logarithm at the point (x, log_{b}(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
 ddxlnf(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
]
The
natural logarithm
of t can be defined as the
definite integral
:
 lnt=∫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 xaxis 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 power formula ln(t^{r}) = 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^{r1},dwright)=rint _{1}^{t}{frac {1}{w}},dw=rln(t).}
The second equality uses a change of variables (
integration by substitution
), w = x^{1/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
]
Logarithms are easy to compute in some cases, such as log_{10}(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 lookup 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)=2log2x.{displaystyle log _{2}left(x^{2}right)=2log _{2}x.}
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Power series[
edit
]
 Taylor series
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 {(z1)^{1}}{1}}{frac {(z1)^{2}}{2}}+{frac {(z1)^{3}}{3}}{frac {(z1)^{4}}{4}}+cdots \&=sum _{k=1}^{infty }(1)^{k+1}{frac {(z1)^{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}(z1)&&\(z1)&&{frac {(z1)^{2}}{2}}&\(z1)&&{frac {(z1)^{2}}{2}}&+&{frac {(z1)^{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 firstorder 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 {z1}{z+1}}=2left({frac {z1}{z+1}}+{frac {1}{3}}{left({frac {z1}{z+1}}right)}^{3}+{frac {1}{5}}{left({frac {z1}{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 {z1}{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 lowaccuracy 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 · 10^{b}, 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 2^{−p} (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^{2m}/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 bitprocessing 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
]
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
selfsimilarity
. 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 selfsimilar 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 largescale 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
]
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
logarithmicscale
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
signaltonoise 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 signaltonoise 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 (10^{1.5}) and a 6.0 releases 1000 times (10^{3}) 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 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 functions
of the form f(x) = a · b^{x} appear as straight lines with
slope
equal to the logarithm of b.
Loglog
graphs scale both axes logarithmically, which causes functions of the form f(x) = a · x^{k} 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
]
Logarithms arise in
probability theory
: the
law of large numbers
dictates that, for a
fair coin
, as the number of cointosses increases to infinity, the observed proportion of heads
approaches onehalf
. The fluctuations of this proportion about onehalf are described by the
law of the iterated logarithm
.^{}
[77]
Logarithms also occur in
lognormal distributions
. When the logarithm of a
random variable
has a
normal distribution
, the variable is said to have a lognormal distribution.^{}
[78]
Lognormal 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
maximumlikelihood 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 parametervalue as a maximum of the logarithm of the likelihood (the “log likelihood“), because the logarithm is an increasing function. The loglikelihood 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 decimaldigit 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.^{}
[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, log_{2}(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 log_{2}(N) + 1
bits
. In other words, the amount of
memory
needed to store N grows logarithmically with N.
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Entropy and chaos[
edit
]
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, 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.^{}
[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
]
Logarithms occur in definitions of the
dimension
of
fractals
.^{}
[91]
Fractals are geometric objects that are
selfsimilar
: 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 logarithmbased notion of dimension is obtained by
counting the number of boxes
needed to cover the fractal in question.
Music[
edit
]
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
Bflat
has a frequency of 466 Hz. The interval between A and Bflat is a
semitone
, as is the one between Bflat 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 base2^{1/12} logarithm of the
frequency
ratio, while the base2^{1/1200} 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 nonequal temperaments.^{}
[92]
Interval (the two tones are played at the same time) 
1/12 tone
play ( help · info ) 
Semitone
play

Just major third
play

Major third
play

Tritone
play

Octave
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
]
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 nonzero 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 2π,{displaystyle 2pi ,} any complex number z may be denoted as
 z=x+iy=r(cosφ+isinφ)=r(cos(φ+2kπ)+isin(φ+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 counterclockwise 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 socalled principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g., −π<φ≤π{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.
Euler’s formula
connects the
trigonometric functions
sine
and
cosine
to the
complex exponential
:
 eiφ=cosφ+isinφ.{displaystyle e^{ivarphi }=cos varphi +isin varphi .}
Using this formula, and again the periodicity, the following identities hold:^{}
[98]
 z=r(cosφ+isinφ)=r(cos(φ+2kπ)+isin(φ+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, a_{k} denote the complex logarithms of z, and k is an arbitrary integer. Therefore, the complex logarithms of z, which are all those complex values a_{k} for which the a_{k}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 a_{k} 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 kvalue 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“,
multivalued 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 (multivalued) inverse function of the
matrix exponential
.^{}
[100]
Another example is the
padic logarithm
, the inverse function of the
padic 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 nonzero elements of a
finite field
.^{}
[104]
Further logarithmlike inverse functions include the double logarithm ln(ln(x)), the
super or hyper4logarithm
(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) = we^{w},^{}
[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 nonnegative 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 oneforms
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 Li_{1}(z) = −ln(1 − z). Moreover, Li_{s}(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
]

^
The restrictions on x and b are explained in the section
“Analytic properties”
.

^
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]

^
For example
C
,
Java
,
Haskell
, and
BASIC
.

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

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

^
See
radian
for the conversion between 2
π
and 360
degree
.
References[
edit
]
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^{a}
^{b}
^{c}
^{d}
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10.1.1.178.3227
,
doi
:
10.1007/9783034886000
,
ISBN
9783764328221
,
MR
1193913
, section 2

^
Apostol, T.M. (2010),
“Logarithm”
, in
Olver, Frank W. J.
; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
NIST Handbook of Mathematical Functions
, Cambridge University Press,
ISBN
9780521192255
,
MR
2723248
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),
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Categories
:

Logarithms

Elementary special functions

Scottish inventions

Additive functions
Chuyên mục: Kiến thức