According to the documentation, Mathematica chooses the branch cut for $\log(z)$ to lie along the negative real axis. It it possible to change this so that it lies along the positive axis or elsewhere in the complex plane?
Let me join the fun and see if I can write my answer without any omissions... both of the previous ones had little errors you can easily check by inverting the newly defined log function
So here is my definition that I just verified:
Note that the sign in front of the branch angle
I think the correct way to achieve this is
Positive or negative θ corresponds to moving up or down the Riemann surface.