Is the Result of Complex Exponent az Multivalued?

[aaaa202]

This is just a quick question which arose when doing an exercise, where you had to evaluate a complex exponent, a^z.
As you know you can easily generalize exponents to complex numbers using the fact that:
a^z = e^{ln(a)$\cdot z$}
However, as you also know the function lnz is multivalued, i.e. ln(rexp(i$\theta)) = ln(r) + i(\theta$+2k$\pi$). Does that mean that the result for a^z should also be multivalued?

 

[micromass]

Yes, the value $a^z$ is multivalued. However, if we take the principal branch of the logarithm, then we get only one value. This principal branch is what is often meant with $a^z$.