 Solution

If a holomorphic function $g$ on the punctured complex plane satisfied $e^{g(z)}=z$, the chain rule would give $e^{g(z)}{g}^{\prime }(z)=1$, hence $g^{\prime }(z)=1/z$. This contradicts the previous part. Therefore there is no global holomorphic logarithm on the punctured plane. A branch of the complex logarithm requires a suitable smaller domain.