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

My attempt at a solution follows:

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