I saw this working recently,
$$\log(z)=\log((1/z)^{-1})=-\log(1/z)$$
After some manipulation, this leads to $\arg(1/z)+\arg(z)=0$. Then, defining $z=re^{i\theta}$, we get that $\theta-\theta=0$ from the above equation. So all seems well. But then after choosing an appropriate branch cut, this does not work with $-1$.
But why? Is it because $-1$ is near a branch cut (if defined in this way)? What if $\arg$ was defined from $0$ to $2\pi$ - then $-1$ is not near the branch cut, but this still does not work.
Q: At what stage in the manipulations did this go wrong? I suppose there is something about complex logarithms that cannot follow the same rules as real logarithms.