For proving the complex logarithm identity $logz_1z_2 = logz_1 + logz_2$, most online resources that I've seen have done this:$$logz_1z_2= ln|z_1z_2| + i*arg(z_1z_2)$$ $$=ln|z_1| + i*arg(z_1) + ln|z_2| + i*arg(z_2)$$
Since $ln|z_1| + i*arg(z_1) = logz_1$ and $ln|z_2| + i*arg(z_2) = logz_2$, $$logz_1z_2 = logz_1 + logz_2$$
But isn't $logz_1 = ln|z_1| + i*arg(z_1) + (2k\pi)i$, where k is any integer? Ideally when k=$0$, shouldn't it reduce to the principal logarithm?
So shouldn't the above identity actually be $logz_1z_2 = Logz_1 + Logz_2$, where $Logz$ denotes the principal logarithm? Where am I going wrong?
$\log (z_1z_2)=\log (z_1)+\log (z_2)$ is not true for any branch of logarithm. Take $z_1=z_2=-1$ to see that this fails for the principal branch.