How to determine the function $\ln z$ for a complex argument $z$?

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On the complex plane, the function $\ln z$ becomes multivalued. If $z=|z| e^{i \varphi}$, then $\ln z$ can be any of the numbers

$w_{k}=\ln |z|+i(\varphi+2 k \pi)(k \in \mathbf{Z})$. For each of them, $e^{w_{k}}=e^{\ln |z|} e^{i(\varphi+2 k \pi)}=|z|(\cos \varphi+i \sin \varphi)=z$.