In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is analytic in the entire plane.
I was a bit surprised by this. My understanding is that, given at point $z_0\neq0$, you can always choose a branch so that log $z$ is analytic at that point. But once you have chosen a branch, that branch will not be analytic in the branch cut. To me this implies, that you can never choose $one$ branch of the logarithm, such that log $f(z)$ is analytic in the whole plane.
Grateful if you can sort this out for me. Where do I go wrong? Am I misinterpreting what is said in the solution?