The natural logarithm, generally written as ln(x), is the logarithm with the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.

The derivative of the natural logarithm is given by

This leads to the Taylor series for ln(1 + x) around 0:

is a picture of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.

Substituting x − 1 for x, we obtain an alternative form for ln(x) itself:

By using Euler transform, we reach the following equation, which is valid for any x with absolute value greater than 1:

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:

In other words,

and

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

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

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts: