Why $\log(1/z) = -\log(z)$ is true for $- \pi < \arg(z) < \pi $ and it's false for $0 < \arg(z) < 2 \pi$?

Question

I don't underestand why this argument is false:

We take $$z = re^{i\theta}.$$

Then $$\log(z^{-1}) = \log(r^{-1}e^{-i\theta}) = -\log(r) - i\theta = -\log(re^{i\theta}) = -\log(z).$$

And why it only works when we take the branch-cut on $- \pi < \arg(z) < \pi $?