What, in set-theoretical terms, is a satisfactory definition of the complex logarithm?

A bijection $\log^+:\Bbb R_{>0}\times\Bbb R\to \Bbb C:(r,\theta)\mapsto\ln r+\mathrm i\theta$ can be defined, along with a family of smooth bijections $\mathrm c_{\alpha}:\Bbb R_{>0}\times (\alpha-\pi\,,\alpha+\pi]\to\Bbb C_{\neq0}:(r,\theta)\mapsto r\mathrm e^{\mathrm i\theta}$ ($\alpha\in\Bbb R$). Then the function $$\log^{(\alpha)}:=\log^+\circ\;\mathrm c_\alpha^{-1}$$maps $\Bbb C_{\neq0}$ injectively to $\Bbb C$, and is continuous on $\Bbb C\setminus\{r\mathrm e^{-\mathrm i\alpha}:r\geqslant0\}$. We could thus define the complex logarithm to be the family $$\log:=\{\log^{(\alpha)}:\alpha\in\Bbb R\}.$$ Any convenient member of this family can then be used as a logarithmic function on $\Bbb C_{\neq0}$ (conventionally with the dependence on $\alpha$ implicit and suppressed in notation).

The above is only my attempt to get a clear picture of the complex logarithm. The definitions I have come across hitherto are simpler than this, but not as clear as I might hope for. What is usually taken to be the simplest and clearest definition of complex logarithm as a set-theoretic object?