Is it true that one can't raise both sides of an equation to a complex power?

Question

There are a lot of counterexamples. Am I missing something? Question: Let $u,v,w \in \mathbb{C}$, does $v=w$ not always imply $v^u=w^u$?

Answer 1

What? $v=w\Rightarrow v^u=w^u$ for all $u,v,w\in \mathbb C$. But it's not true that $v^u=w^u\implies v=w$, thats because the map $(-)^u$ is not alway injective, not only in $\mathbb C$ but also in $\mathbb R$! (i.e. if u=2 then (-1)^2=(1)^2 )