Let $D\subset \mathbb C$ be an arbitrary domain not containing $0$. Assume that the function $\frac{1}{z}$ has a holomorphic primitive $F$ on $D$. Show that for some constant $c \in \mathbb C$:$$F(z)+c \in \operatorname{Ln}(z)\ \ \forall z \in D.$$
To my knowledge the function $1/z$ has a holomorphic primitive $\ln|z|+i\operatorname{arg}(z)$ on $\mathbb C\setminus{\mathbb R\_+} $. But how can I construct something similar for a unknown domain? Any help is much appreciated.