[Math] Branch points and Branch cut of $f(z) = \log(z^2)$

Question

I am trying to find the branch points and choose a branch cut for the function
$f(z) = \log(z^2)$.

I know that both $z = 0$ and $z = \infty$ are branch points, so it seems reasonable to just choose the negative real axis as a branch cut as with $\log{z}$. However, as $\theta$ goes from $0$ to $2\pi$, $\arg{z^2}$ goes from $0$ to $4\pi$ so I am not sure whether the branch cut stated above is sufficient.

My understanding of all this is still not very strong, so any tips or intuition are appreciated. Are the two branch points and the branch cut stated above correct?

Answer 1

Note that $z$ must satisfy that $z^2 \not \in (-\infty,0)$ taking the usual cut. What values of $z$ make that happen?