Solve equation with complex numbers

Question

suppose I have some equation of this form $A(z)=B(z)$. Is valid to just solve for $\log(A(z))=\log(B(z))$ or $\exp(A(z))=\exp(B(z))$ without caring what exactly $A$ and $B$ are or the domain of $z$? By valid, I mean not losing or acquiring more solutions.

Answer 1

$$\exp(w+i2\pi)=\exp(w),$$ so taking the exponential introduces extra solutions.

And taking the logarithm doesn't work for zero arguments.