I'm having trouble understand branch cuts. I understand that a function is discontinuous along their branch cut but what I don't understand is how you can change the branch cut in order to solve equations.

Example: determine a branch of f(z) = log(z^3-2) that is analytic at z=0 and find f(0) and f '(0).

Here, the inside is -2 when z=0, so you'd choose a branch which isn't -Pi like -Pi/4. so f(0)=Log|-2| + Arg (-2). I guess my question is how do you find Arg(-2) with different branch cuts.