It is better in that you eliminate the
's in the final answer but I would still prefer a solution that did not have to worry about which branch of the
was in use where. That can be done by the the first step that I proposed (which, while that is not the way I think of it, absorbs the taking of logarithms while effectively fixing the branch).
Also as
is multi-valued the right hand side of final line is just one of the solutions that correspond to the left hand side (as you know, and I know you know).
CB