Definition:Natural Logarithm/Complex

Definition
Let $z = r e^{i \theta}$ be a complex number expressed in exponential form.

The complex natural logarithm of $z \in \C$ is:


 * $\log \left({z}\right) := \ln r + i \theta + 2 k \pi i$

and is a multifunction.

Also defined as
The complex natural logarithm of a complex value $z \in \C$ is written $\ln \left({z}\right)$ and is defined as:


 * $\ln \left({z}\right) := \ln \left|{z}\right| + i \arg \left({z}\right)$

where $\arg \left({z}\right)$ is the continuous argument of $z$ and $\operatorname{Arg}\left({z}\right) = \arg \left({z}\right) \ \left({\bmod \left({2 \pi}\right)}\right)$ is the principal argument of $z$.

The principal branch of the complex logarithm is written and defined:


 * $\operatorname{Log} \left({z}\right) := \ln \left|{z}\right| + i \operatorname{Arg} \left({z}\right)$

Notation
The natural logarithm of $x$ is written variously as:


 * $\ln z$
 * $\log z$
 * $\log_e z$

The first of these is fairly commonly encountered, and frequently preferred. The second is ambiguous (it doesn't tell you which base it is the logarithm of) and the third is verbose.

However, notation is misleadingly inconsistent throughout the literature, and if there is any confusion about exactly what is meant, the full (verbose) format can be argued for.

Note on Definition
For any definition of the exponential function that is not "the function inverse of the natural logarithm", the natural logarithm can be defined as the inverse of the exponential function.

Also see

 * Properties of Natural Logarithm