Take a set $A \subseteq \mathbb{C}$ so that the function $exp: A \to \mathbb{C}_{0}$ is a bijection. Let $L_{A}: \mathbb{C}_{0} \to A$ be the inverse of $exp$.
The question is if $L_{A}(z_{1}z_{2})= L_{A}(z_{1})+L_{A}(z_{2})$ will hold for every $z_{1},z_{2} \in \mathbb{C}_{0}$. Of course this is true in $\mathbb{R}$ and I also think that this holds for the complex numbers but I can't find a proof. Is there anyone that can help me?
I presume $\Bbb C_0=\Bbb C^*$, the set of non-zero complexes.
Then an allowable $A$ is $\{x+iy:0\le y<2\pi\}$. What about $z_1=z_2=-1$?