---
tag: 0X2FY491PF
kind: definition
name: Complex logarithm
msc: [33B10]
defines: 02QGGD5G87
uses: [054VXHMCA3, 0ZZRSX91QR, 0KW2X0TFBX]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_logarithm
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Complex_logarithm
    locator: Problems with inverting the complex exponential function
---

## Definition

Let $z$ be a complex number with $z \neq 0$.
A complex logarithm of $z$ is a complex number $w$ with $e^w = z$, where $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.
The notation $\log z$ means the set of all complex logarithms of $z$, or one chosen element of this set.

## Remarks

Let $|z|$ be the [[modulus|0KW2X0TFBX]] of $z$, let $\ln$ be the real natural logarithm, and let $\theta$ be an [[argument|0ZZRSX91QR]] of $z$. Then the logarithms of $z$ are exactly the numbers $\ln |z| + i (\theta + 2 \pi k)$ with $k \in \mathbb{Z}$.
The number $0$ has no logarithm, because $e^w \neq 0$ for each $w$.
So $\log$ is a [[multivalued function|03CPD1HGF3]] on $\mathbb{C} \setminus \{0\}$. Its [[principal branch|0YKY4M3KRQ]] is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, with $-\pi < \operatorname{Arg} z \le \pi$.
For example, the logarithms of $-1$ are $i \pi (2k + 1)$ with $k \in \mathbb{Z}$, and $\operatorname{Log}(-1) = i \pi$.

## Explanations

### L1 human

A logarithm of $z$ is an answer to the question: $e$ to which power gives $z$?
Because $e^{w + 2\pi i} = e^w$, each answer gives infinitely many other answers, spaced by $2\pi i$ on a vertical line.
The real part of each answer is $\ln |z|$, and the imaginary part is an angle of $z$.