I didn't really understand why the multi-valued function, $\log$ was used instead of single-valued function, $\mathrm{Log}$.
Also, could you please explain me how in the examples, the $\Delta{\arg f(z)}$ is $6\pi$, $-2\pi$, and $0$?
I don't understand because the why in 1. and 2. would actually count the total argument changed when $\theta$ is rotating from $0$ to $2\pi$, whereas 3. would just result $0$, not $2\pi$.
It would be great to know how to calculate $\Delta{\arg f(z)}$.