The

← Logarithm

Definitions

The general (multivalued) logarithm $\mathrm{Ln}z$ is defined as $$\mathrm{Ln}z={\int }_1^z\frac{1}{t}dt,z\ne 0,$$ with the integration path not intersecting the origin.

Its principal value $\ln z$ is defined as $$\ln z={\int }_1^z\frac{1}{t}dt,z\notin (-\infty ,0],$$ with the integration path not intersecting $(-\infty ,0]$. The principal value is extended to include the branch cut $z\in (-\infty ,0)$ such that it is continuous on the upper half plane $\mathrm{Im}z\ge 0,z\ne 0$.

The logarithm to a general base $a$ is defined as $${\log }_{a}z=\frac{\ln z}{\ln a}.$$

Fundamental Properties

The general logarithm can be defined as the inverse of the exponential function, $$\exp z=w\iff z=\mathrm{Ln}w.$$ The imaginary part of the principal value satisfies $$\mathrm{Im}\ln z\in (-\pi ,\pi ].$$ The principal value is obtained by restricting the general logarithm to this region. Conversely, the general logarithm is obtained from the principal value via $$\mathrm{Ln}z=\ln z+2\pi ik,$$ where $k$ is the number of times the integration path in $\mathrm{Ln}z$ crosses the negative real axis from above, minus the number of times it crosses from below.

The real and imaginary parts of $\ln z$ are $$\mathrm{Re}\ln z=\ln |z|,$$ $$\mathrm{Im}\ln z=\arg z,$$ where $\arg z$ is the argument of $z$ satisfying $\arg z\in (-\pi ,\pi ]$.

Identities

For complex numbers $a,b,z$ and integer $n$ the following holds: $$\begin{array}{rl}\exp \ln z& =\exp \mathrm{Ln}z=z\\ \ln \exp z& =z,& & \text{if }\mathrm{Im}z\in (-\pi ,\pi ]\\ \ln z^n& =n\ln z,& & \text{if }\arg z\in (-\pi ,\pi ]\\ \ln (ab)& =\ln a+\ln b,& & \text{if }\arg a+\arg b\in (-\pi ,\pi ]\\ \ln (a/b)& =\ln a-\ln b,& & \text{if }\arg a-\arg b\in (-\pi ,\pi ]\end{array}$$

The logarithm of Euler's number $e$ is equal to one: $$\ln e=1.$$