Is this log identity true?

Question

I'm wondering if the exponent property carries forward to the complex log. In other words, for some complex numbers $z$ and $w$ does $\ln(z^w) = w\ln(z)$?

Answer 1

You have to be careful with branches of $\ln$. The definition of $z^w$ for complex $z$ and $w$ is $z^w = \exp(w \ln(z))$ (for some branch of $\ln$). Therefore $\ln(z^w) = w \ln(z) + 2 \pi i n$ where $n$ is an integer.
Which integer it is will depend on what branch of $\ln$ you want to use.