Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is approximately equal to 2.718281828459…. The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers as will be explained below.

Graph of the natural logarithm function. The function goes to negative infinity as x approaches 0, but grows slowly to positive infinity as x increases in value.

The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:

$e^{\ln(x)}=x\qquad {\mbox{if }}x>0\,\!$

$\ln(e^{x})=x.\,\!$

In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition.

Logarithms can be defined to any positive base other than 1, not just e, and are useful for solving equations in which the unknown appears as the exponent of some other quantity.

Notational conventions

• Mathematicians generally understand either "log(x)" or "ln(x)" to mean log_e(x), i.e., the natural logarithm of x, and write "log₁₀(x)" if the base-10 logarithm of x is intended. However, often in the United Kingdom, "lg(x)" and/or "log(x)" with no base specified is taken to mean "log₁₀(x)".

• Engineers, biologists, and some others write only "ln(x)" (or occasionally "log_e(x)") when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x) or, in the context of computing, log₂(x).

• In most commonly-used programming languages, including C, C++, Fortran, and BASIC, "log" or "LOG" refers to the natural logarithm.

• On hand-held calculators, the natural logarithm is denoted ln, whereas log is the base-10 logarithm.

Reason for being "natural"

Initially, it seems that in a world using base 10 for nearly all calculations, this base would be more "natural" than base e. The reason we call the ln(x) "natural" is twofold: first, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10, and second, because the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms. Thus, the natural logarithm is more useful in practice. To put it concretely, consider the problem of differentiating a logarithmic function:

${\frac {d}{dx}}\log _{b}(x)={\frac {\log _{b}e}{x}}$

If the base b is equal to e then the derivative is simply 1/x, and at x = 1 the slope of the graph is 1.

There are other reasons the natural logarithm is natural; there are a number of simple series involving the natural logarithm, and it often arises in nature. In fact, Nicholas Mercator first described them as log naturalis before calculus was even conceived.

Definitions

Formally, ln(a) may be defined as the area under the graph (integral) of 1/x from 1 to a, that is,

$\ln(a)=\int _{1}^{a}{\frac {1}{x}}\,dx.$

This defines a logarithm because it satisfies the fundamental property of a logarithm:

$\ln(ab)=\ln(a)+\ln(b)\,\!$

This can be demonstrated by letting $t={\tfrac {x}{a}}$ as follows:

$\ln(ab)=\int _{1}^{ab}{\frac {1}{x}}\;dx=\int _{1}^{a}{\frac {1}{x}}\;dx\;+\int _{a}^{ab}{\frac {1}{x}}\;dx=\int _{1}^{a}{\frac {1}{x}}\;dx\;+\int _{1}^{b}{\frac {1}{t}}\;dt=\ln(a)+\ln(b)$

The number e can then be defined as the unique real number a such that ln(a) = 1.

Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, i.e., ln(x) is that function such that $e^{\ln(x)}=x\!$. Since the range of the exponential function on real arguments is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive x.

Derivative, Taylor series and complex arguments

The derivative of the natural logarithm is given by

${\frac {d}{dx}}\ln(x)={\frac {1}{x}}.\,$

This leads to the Taylor series

$\ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots \quad {\rm {for}}\quad \left|x\right|\leq 1\quad {\rm {unless}}\quad x=-1,$

which is also known as the Mercator series.

Substituting x-1 for x, we obtain an alternative form for ln(x) itself, namely

$\ln(x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}(x-1)^{n}=x-1-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-{\frac {(x-1)^{4}}{4}}\cdots$

${\rm {for}}\quad \left|x-1\right|\leq 1\quad {\rm {unless}}\quad x=0.$

The natural logarithm in integration

The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact:

$\ {d \over dx}\left(\ln \left|x\right|\right)={1 \over x}.$

In other words,

$\int {dx \over x}=\ln |x|+C$

and

$\int {{\frac {f'(x)}{f(x)}}\,dx}=\ln |f(x)|+C.$

Here is an example in the case of g(x) = tan(x):

$\int \tan(x)\,dx=\int {\sin(x) \over \cos(x)}\,dx$

$\int \tan(x)\,dx=\int {-{d \over dx}\cos(x) \over {\cos(x)}}\,dx.$

Letting f(x) = cos(x) and f'(x)= - sin(x):

$\int \tan(x)\,dx=-\ln {\left|\cos(x)\right|}+C$

$\int \tan(x)\,dx=\ln {\left|\sec(x)\right|}+C$

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts:

$\int \ln(x)\,dx=x\ln(x)-x+C.$

Numerical value

To calculate the numerical value of the natural logarithm of a number, the Taylor series expansion can be rewritten as:

$\ln(1+x)=x\,\left({\frac {1}{1}}-x\,\left({\frac {1}{2}}-x\,\left({\frac {1}{3}}-x\,\left({\frac {1}{4}}-x\,\left({\frac {1}{5}}-\ldots \right)\right)\right)\right)\right)\quad {\rm {for}}\quad \left|x\right|<1.\,\!$

To obtain a better rate of convergence, the following identity can be used.

$\ln(x)=\ln \left({\frac {1+y}{1-y}}\right)$	$=2\,y\,\left({\frac {1}{1}}+{\frac {1}{3}}y^{2}+{\frac {1}{5}}y^{4}+{\frac {1}{7}}y^{6}+{\frac {1}{9}}y^{8}+\ldots \right)$
	$=2\,y\,\left({\frac {1}{1}}+y^{2}\,\left({\frac {1}{3}}+y^{2}\,\left({\frac {1}{5}}+y^{2}\,\left({\frac {1}{7}}+y^{2}\,\left({\frac {1}{9}}+\ldots \right)\right)\right)\right)\right)$

provided that y = (x−1)/(x+1) and x > 0.

For ln(x) where x > 1, the closer the value of x is to 1, the faster the rate of convergence. The identities associated with the logarithm can be leveraged to exploit this:

$\ln(123.456)\!$	$=\ln(1.23456\times 10^{2})\,\!$
	$=\ln(1.23456)+\ln(10^{2})\,\!$
	$=\ln(1.23456)+2\times \ln(10)\,\!$
	$\approx \ln(1.23456)+2\times 2.3025851\,\!$

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

High precision

To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. An alternative is to use Newton's method to invert the exponential function, whose series converges more quickly.

An alternative for extremely high precision calculation is the formula

$\ln x\approx {\frac {\pi }{2M\left(1,{\frac {4}{s}}\right)}}-m\log 2$

where M denotes the arithmetic-geometric mean and

$s=x\,2^{m}>2^{\frac {p}{2}},$

with m chosen so that p bits of precision is attained. In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.)

Computational complexity

Main article: Computational complexity of mathematical operations

The computational complexity of computing the natural logarithm (using the arithmetic-geometric mean) is O(M(n) ln n). Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M(n) is the computational complexity of multiplying two n-digit numbers.

Complex logarithms

Main article: Complex logarithm

The exponential function can be extended to a function which gives a complex number as e^x for any arbitrary complex number x; simply use the infinite series with x complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no x has e^x = 0; and it turns out that e^2πi = 1 = e⁰. Since the multiplicative property still works for the complex exponential function, e^z = e^z+2nπi, for all complex z and integral n.

So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding 2πi at will. The complex logarithm can only be single-valued on the cut plane. For example, ln i = 1/2 πi or 5/2 πi or −3/2 πi, etc.; and although i⁴ = 1, 4 log i can be defined as 2πi, or 10πi or −6 πi, and so on.

• Plots of the natural logarithm function on the complex plane (principal branch)
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  z = Re(ln(x+iy))
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See also

• John Napier
• Logarithmic integral function
• Nicholas Mercator
• Polylogarithm
• Von Mangoldt function