Confirm that the law of exponents $\displaystyle z^{\lambda}z^{\mu} = z^{\lambda+\mu}$ is valid for all non-zero complex numbers $\displaystyle z$ and all complex exponents $\displaystyle \lambda$ and $\displaystyle \mu$. Give an example of complex numbers $\displaystyle z, \lambda$ and $\displaystyle \mu$ for which $\displaystyle z^{\lambda\mu} \neq z^{\lambda\mu}$.

My attempt at a solution follows:

$\displaystyle z^{\lambda} z^{\mu} = e^{\lambda Log(z)} e^{\mu Log(z)}$

$\displaystyle =e^{\lambda[ Log(z) + i Arg(z)]} + e^{\mu[Log(z)+i Arg(z)]}$

$\displaystyle =e^{(\lambda+\mu)[Log(z) + i Arg(z)]}$

$\displaystyle =e^{\lambda+\mu Log(z)}$

$\displaystyle =z^{\lambda+\mu}$.

I am fairly certain this is at least on the way to being right, but I have no idea about a counterexample. Will you please look over it and see if I'm on the right track and then perhaps point me in the direction to developing a counterexample? Thanks.