Complex logarithm

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 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 of $z$, let $\ln$ be the real natural logarithm, and let $\theta$ be an argument 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 on $\mathbb{C} \setminus \{0\}$. Its principal branch 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 $z$.