The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2.718281828459... (continuing infinitely). 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.

Notational conventions

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• 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.

• 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" means natural logarithm.

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

See also logarithms.

The natural logarithm is the inverse of the natural exponential function

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This function is the inverse function of the exponential function:

$\backslash \; e^\{\backslash log(x)\}\; =\; x\; \backslash ,\backslash !$      for all positive x and

$\backslash log(e^x)\; =\; x\; \backslash ,\backslash !$      for all real 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 they are always useful for solving equations in which the unknown appears as the exponent of some other quantity.

Reason for being "natural"

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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 log(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 (because of the "natural" properties of the exponential function which allow it to describe growth and decay behaviors), 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:

$\backslash frac\{d\}\{dx\}\backslash log\_b(x)\; =\backslash frac\{1\}\{x\; \backslash cdot\; \backslash log\; b\}$

If the base (b) is e then the derivative is 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. Indeed, Nicholas Mercator first described them as log naturalis before calculus was even conceived.

Definitions

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

$\backslash log(a)=\backslash int\_1^a\; \backslash frac\{1\}\{x\}\backslash ,dx.$

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

$\backslash log(ab)=\backslash log(a)+\backslash log(b)\; \backslash ,\backslash !$

This can be shown by defining $\backslash phi(t)=at$ and using the substitution rule of integration as follows:

$$

\log (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 = \log (a) + \log (b)

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

Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, meaning log(x) is that number for which $e^\{\backslash log(x)\}\; =\; x$ Since the range of the exponential function 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

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The derivative of the natural logarithm is given by

$\backslash frac\{d\}\{dx\}\; \backslash log(x)\; =\; \backslash frac\{1\}\{x\}.\backslash ,$

This leads to the Taylor series

which is also known as the Mercator series.

An alternative form "Logarithmic Expansions" at Math2.org for log(x) itself is

The natural logarithm in integration

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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:

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

In other words,

$\backslash int\; \{\; dx\; \backslash over\; x\}\; =\; \backslash log|x|\; +\; C$

and

$\backslash int\; \{\; \{f^\text{'}(x)\; \backslash over\; f(x)\}\backslash ,\; dx\}\; =\; \backslash log\; |f(x)|\; +\; C.$

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

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

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

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

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

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

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts:

$\backslash int\; \backslash log\; (x)\; \backslash ,dx\; =\; x\; \backslash log\; (x)\; -\; x\; +\; C.$

Numerical value

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

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

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

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

For log(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:

{| $\backslash log(123.456)$ $=\; \backslash log(1.23456\; \backslash times\; 10^2)\; \backslash ,\backslash !$ $=\; \backslash log(1.23456)\; +\; \backslash log(10^2)\; \backslash ,\backslash !$ $=\; \backslash log(1.23456)\; +\; 2\; \backslash times\; \backslash log(10)\; \backslash ,\backslash !$ $\backslash approx\; \backslash log(1.23456)\; +\; 2\; \backslash times\; 2.3025851\; \backslash ,\backslash !$

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

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

where M denotes the arithmetic-geometric mean and

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

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.)

See also

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• Complex logarithm
• John Napier
• Logarithmic integral function
• Von Mangoldt function

References

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Logarithms | Elementary special functions

Натурален логаритъм | Naturlig logaritme | Logarithme naturel | 자연로그 | Logaritmo naturale | לוגריתם טבעי | Natūrinis logaritmas | Natuurlijke logaritme | 自然対数 | Logarytm naturalny | Logaritmo natural | Natural logarithm | ลอการิทึมธรรมชาติ | 自然對數