I'm having a bit of trouble proving the following property:
Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch.
I know that $\log (zw) = \log(z) + \log(w) \space mod \space 2\pi i$, but I am not really sure how that helps. Any thoughts?
Hint: Assume neither of $z,w$ is $0.$ We can write $z=re^{ia}, w = se^{ib},$ with $a\in (-\pi/2,\pi/2), b\in [-\pi/2,\pi/2].$ Then $zw = (rs)e^{i(a+b)}$ and $a+b \in (-\pi,\pi).$