Suppose $f: G\rightarrow \mathbb{C}$ is a branch of $z^a$. Then $f(z) = e^{ah_1(z)}$ for all $z\in G$. Likewise if $g:G\rightarrow \mathbb{C}$ is a branch of $z^b$ then $g(z) = e^{bh_2(z)}$ for all $z\in G$.
Here $h_1$ and $h_2$ are branches of the logarithm on $G$, and hence they differ by $2\pi k i$ with $k\in \mathbb{Z}$.
However then $fg(z) = f(z)g(z) = e^{ah_1}\cdot e^{b(h_1+2\pi k i)}= e^{(a+b)h_1+2\pi k b i}$, which is not a branch of $z^{a+b}$.
Question: Should I be using the same branch of $\log$ when defining the branches of $z^a$ and $z^b$. That is, should I write $g(z) = e^{bh_1(z)}$, beacause then everything works out. Also, why should I be choosing the same $\log$. Thanks.