Prove that the principal logarithm ($Log$) is a bijection between $\mathbb{C}-[0,\infty)$ and $\Omega=\{z \in \mathbb{C} | -\pi < Im(z) <\pi\}$

Question

I'm doing this exercise:

Prove that the principal logarithm ($Log$) is a bijection between $\mathbb{C}-[0,\infty)$ and $\Omega=\{z\in\mathbb{C}|-\pi<Im(z)<\pi\}$

First of all, I understand that $\mathbb{C}-[0,\infty)$ is referring to the set of all complex numbers minus the positive real axis, right? Is it the notation we have to use?

When we have this solved, I have to see that $Log$ is injective ($Log(a)=Log(b) \iff a=b$) and that is exhaustive. Then, we're done, right?

I'd like some hints because I'm confused by this problem's notation.

Thanks for your time.

Answer 1

Hint: use polar coordinates. With some caveats $$\log(r e^{i\theta}) = \log r + i\theta.$$