For what subset of $\mathbb{C}$ is $ $ Log$(\alpha^z) =z$Log$(\alpha) \ $ true?

Question

For what subset of $\mathbb{C}$ is the following true?

Log$(\alpha^z) =z$Log$(\alpha)$

where Log is the principal log, $z \ $ a complex variable and $\alpha$ a nonzero complex constant.

EDIT (it seems my misconception is deep as the below actually seem false)
I know that when the roles of $\alpha$ and $z$ are swapped, the equality holds. i.e.

Log$(z^{\alpha}) =\alpha$Log$(z),\ $ for $ \ z\in \mathbb{C}^*$,

is true.

I'm not so sure for the first.

Answer 1

Suppose that $\alpha=-1+i$ and that $z=2$. Then:

• $\operatorname{Log}(\alpha^z)=\operatorname{Log}(-2i)=-\frac\pi2$;
• $z\operatorname{Log}(\alpha)=2\times\frac{3\pi}4=\frac\pi2$.

Therefore, $\operatorname{Log}(\alpha^z)\neq z\operatorname{Log}(\alpha)$.