TITLE: Law of exponent for complex numbers
QUESTION [7 upvotes]: Why does the following law
$$(z_1z_2)^w=z_1^wz_2^w$$
sometimes fail, such as when $z_1=z_2=-1$ and $w=-i$?
And what are the conditions I need for $z_1$ and $z_2$ so that the above identity hold for all complex values of w?

REPLY [2 votes]: Note first that when $w=0$, then clearly the equality holds. So let $w \neq 0$. Then
$$(z_1z_2)^w = z_1^wz_2^w \iff exp(wln(z_1z_2)) = exp(wln(z_1))exp(wln(z_2)) \iff exp(wln(z_1z_2)) = exp(w(ln(z_1)+ln(z_2))) \iff wln(z_1z_2) = w(ln(z_1)+ln(z_2)) \iff ln(z_1z_2) = ln(z_1)+ln(z_2)$$
Hence we found the required condition on $z_1$ and $z_2$ such that the equality holds, namely 
$$ln(z_1z_2) = ln(z_1)+ln(z_2)$$