This was a problem from Freitag's Complex Analysis,
A continuous function $l: D \rightarrow \mathbb{C}$, $D$ open connected subset of $\mathbb{C} \setminus \{0 \}$, is called a continuous branch of log if $\exp ^{l(z)}= z $ for all $z \in D$.
Show: On $D$ there exists a unique continuous branch of the logarithm only if the function $1/z$ has primitive on $D$.
I know we can work locally: if $l$ is primitive, and $l'=1/z$, then on $\bar{B}(a,\epsilon)$ take an analytic branch $L_\varepsilon$ defined on this domain, we deduce $\exp^{l(z)}=z $ for all $z \in B(a,\varepsilon)$.
So $l$ is in fact a continuous branch.
This seems a bit counter intuitive. Wouldn't any $\bar{l} = l +2 \pi k i$ be another continuous branch on $D$? Then why uniqueness?