The

**natural logarithm**is the logarithm to the base

*e*, where

*e*is approximately equal to 2.71828... (no exact fraction can be given, as

*e*is an irrational number). The natural logarithm is defined for all positive real numbers

*x*and can also be defined for non-zero complex numbers as will be explained below. Although this function was not introduced by Napier, it is sometimes known as the

**Naperian Logarithm**.

ln(x)

## Notational conventions

Most of the reason for thinking about base-10 logarithms became obsolete shortly after about 1970 when hand-held calculators became widespread (for more on this point, see common logarithm). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(*x*)" to mean the base-10 logarithm of *x* and use only "ln(*x*)" to refer to the natural logarithm of *x*. As recently as 1984, Paul Halmos in his autobiography heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) At the time of this writing (2003), many mathematicians have adopted the "ln" notation, but "log" also remains in widespread use.

To avoid all confusion, Wikipedia uses the notation ln(*x*) for the natural logarithm of *x* and log_{10}(*x*) for the base-10 logarithm of *x*.

## Ln is the inverse of the natural exponential function

This function is the inverse function of the exponential function, thus it holds

*e*^{ln(x)}=*x*for all positive*x*and- ln(
*e*^{x}) =*x*for all real*x*.

*e*, and they are always useful for solving equations in which the unknown appears as the exponent of some other quantity.

## What's so "natural" about them?

Initially, it seems that the base-10 would be more "natural" than base *e*. The reason we call ln(*x*) "natural" is twofold: first, the natural logarithm can be defined quite easily using a simple integral or Taylor series as will be explained below; this is not true of other logarithms. Second, 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 to describe growth and decay behaviors), and so the natural logarithm is more useful in practice.
To put it concretely, consider the problem of differentiating a logarithmic function:

*e*is the "constant" equal to 1.

## Definitions

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

*φ(t)*=

*at*and using the substitution rule of integration as follows:

*e*can then defined as the unique real number with ln(

*e*) = 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 ln(*x*) is that number for which *e*^{ln(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

The derivative of the natural logarithm is given by

*z*) also for all non-zero complex numbers

*z*. The above Taylor expansion remains valid for all

*complex*numbers

*x*with absolute value less than 1. If the non-zero complex number

*z*is expressed in polar coordinates as

*z*=

*r*

*e*

^{iφ}with

*r*> 0 and -&pi < φ ≤ +π, then

- ln(
*z*) = ln(*r*) +*i*φ

*e*^{ln(z)}=*z*for all nonzero*z*

*e*

^{z}) does not always equal

*z*, and ln(

*zw*) does not always equal ln(

*z*) + ln(

*w*).

A somewhat more natural definition of ln(*z*) interprets it as a multi-valued function: for *z* = *r* *e*^{iφ} we set

- ln(
*z*) = { ln(*r*) +*i*(φ + 2π*k*) :*k*any integer }

*all*complex numbers

*u*for which

*e*

^{u}=

*z*, because

*e*

^{2πi}= 1 (see The most remarkable formula in the world).

The preferred way to deal with multivalued functions like this in complex analysis is via Riemann surfaces: the function ln is then non defined on the complex plane but instead on a suitable Riemann surface having countably many "leaves" and the values of the function differ by 2π*i* from leaf to leaf.

## Ln integration condition

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:

*g*(

*x*) = tan(

*x*):

*f*(

*x*) = cos(

*x*) and f'(

*x*)= - sin(

*x*):

*C*is an arbitrary constant of integration.

Here is the natural logarithm integrated:

- \n