Branch of logarithm and composition.

Question

Let $\log:G\to\mathbb{C}$ be a branch of logarithm and $f,g:G\to\mathbb{C}$ the corresponding branches of $z^a$ and $z^b$ respectively.

I need an example to show that even with $f(G)\subset G$, $g \circ f$ is not a branch of $z^{ab}$.