The complex logarithm defines a relation
\[ \log \colon \mathbb {C}\to \mathcal{P}\webleft (\mathbb {C}\webright ) \]
from $\mathbb {C}$ to itself, where we have
\[ \log \webleft (a+bi\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \log \webleft (\sqrt{a^{2}+b^{2}}\webright )+i\arg \webleft (a+bi\webright )+\webleft (2\pi i\webright )k\ \middle |\ k\in \mathbb {Z}\webright\} \]
for each $a+bi\in \mathbb {C}$.
