Difference between branches $[-\pi, \pi)$ and $[\pi, 3\pi)$ of the complex logarithm

Question

I think that both branches just exclude the negative part of the real line. So what's the difference between them then?

Answer 1

For $x \in \mathbf{R}$, the numbers $x$ and $x+2\pi$ can be said to represent the same angle, but they are still different numbers. So when computing the logarithm of a given complex number using the two branches you describe, you obtain two different complex numbers (which differ by $2\pi i$).