Is it possible for two complex numbers to have a real exponentiation?

In other words, if `x` and `y` are complex (each have the form `a+b`*i*), show that `x^y` can have a real value, or prove that it is impossible.

**Note**: *i* is the imaginary value defined as the number that yields `-1` when squared. `a` and `b` are any real numbers, but `b` is not 0.

Some bits of notation. Let R\P be polar notation for R(cos(P)+i.sen(P)). Let ^ denote exponentiation.

By definition, X^Y = e^(Y.ln(X)). If the imaginary part of Y.ln(X) is zero, the result is real.

Let X=R\P; then ln(X)=ln(R)+i.P. Also, let Y=R'\P' = R'(cos(P')+i.sen(P')). The imaginary part of Y.ln(X) is then R'(ln(R).sen(P')+P.cos(P'))=0.

If we don't allow R'=0, we could take

P=-1/cos(P'), ln(R)=1/sen(P'), and we would have the sought result.

Did I miss anything? Opinions?