The natural logarithm is the logarithm having base e, where
This function can be defined
This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,
The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas denotes the common logarithm.
There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use (i.e., using a trigonometric function-like convention), it is also common to write .
Common and natural logarithms can be expressed in terms of each other as
The natural logarithm is especially useful in calculusbecause its derivative is given by the simple equation
whereas logarithms in other bases have the more complicated derivative
The natural logarithm can be analytically continued to complexnumbers as
where is the complex modulus and is the complex argument. The natural logarithm is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at .
The principal value of the natural logarithm is implemented in the Wolfram Language as Log[x], which is equivalent to Log[E, x]. This function is illustrated above in the complex plane.
Note that the inverse trigonometric and inverse hyperbolic functions can be expressed (and, in fact, are commonly defined) in terms of the natural logarithm, as summarized in the table below. Therefore, once these definition are agreed upon, the branch cut structure adopted for the natural logarithm fixes the branch cuts of these functions.
|inverse hyperbolic cosecant|
|inverse hyperbolic cosine|
|inverse hyperbolic cotangent|
|inverse hyperbolic secant|
|inverse hyperbolic sine|
|inverse hyperbolic tangent|
The Mercator series
gives a Taylor series for the natural logarithm.
Continued fraction representations of logarithmicfunctions include
(Lambert 1770; Lagrange 1776; Olds 1963, p. 138; Wall 1948, p. 342) and
(Euler 1813-1814; Wall 1948, p. 343; Olds 1963, p. 139).
For a complex number , the natural logarithm satisfies
where is the principal value.
Some special values of the natural logarithm include
Natural logarithms can sometimes be written as a sum or difference of "simpler" logarithms, for example
which follows immediately from the identity
Plouffe (2006) found the following beautiful identities: