Suppose I have a branch of the logarithm, that is, a continuous function $L(z)$ on some region $\Omega$ such that $e^{L(z)} = z$. We see that this defines a branch for the square root function on $\Omega$, via $\sqrt{z} = \exp(1/2 L(z))$, since
$(\exp(1/2 L(z))^2 = \exp(L(z)) = z$
I am wondering if a sort of converse of this holds. Suppose on the other hand, we have a branch for the square root, i.e. some continuous function $R(z)$ on $\Omega$ such that $R(z)^2 = z$. Is there some way to get a branch of the logarithm from $R(z)$? If so, does this generalize (i.e. what branches for multi-valued functions will determine a branch of the logarithm)?