TITLE: Is there a reason why $\text{Arg}(z)$ behaves like a logarithm?
QUESTION [6 upvotes]: In school, we learnt how to prove some of the basic properties of $\text{Arg}$, one of them being
$$
\text{Arg}(z_1)+\text{Arg}(z_2)=\text{Arg}(z_1z_2)
$$
We did this by writing $z_1$ and $z_2$ in the modulus-argument form, and using $e^{i\theta}=\cos\theta+i\sin\theta$. I noticed that $\text{Arg}(z)$ has a similar property to logarithms, in that
$$
\log a+\log b = \log ab
$$
Is this just a coincidence, or is there a deeper reason for this?

REPLY [4 votes]: The imaginary part of $\log z$ is $\text{arg}$ z.
$$\log z = \log |z| + i \,\text{arg } z$$