$\log: \mathbb{C}\setminus\{0\} \to C$ defined as
$$z=re^{i \phi} \mapsto \ln (r)+i\phi$$ with $\phi \in [0, 2\pi)$.
I want to show that $\log$ is measurable, that is that for every borel-set $B$ in $\mathbb{C} \simeq \mathbb{R^2}$, $\log^{-1}(B)$ is a borel-set as well.
The hint we received is that we should use that $\exp$ is injective and continuous on $[-n,n]\times[0,2\pi-\frac{1}{n}]$ but I don't know how that should help.
I know that $\log$ is continuous on $\mathbb{C} \setminus[0,\infty)$ and therefore measurable but I need to show it for $\mathbb{C} \setminus \{0\}$
Note that $\mathbb C \setminus \{0\}$ is the disjoint union of the Borel sets $\mathbb C\setminus [0,\infty)$ and $(0,\infty)$. Your branch of the logarithm is continuous, hence measurable, when restricted to each of these sets. It follows that this branch of the logarithm is measurable on $\mathbb C \setminus \{0\}.$