Geometrically, what does raising a real number to a complex number do on the complex plane?

I know the formula to raise a real number to a complex number:
$a^{b+ic}=a^b\cdot(\cos(b \ln a) + i\sin(b \ln a))$
but I don't understand how it's derived. I know what the trig functions are doing but I'm not quite understanding what the natural log is doing and how this relationship transforms on the plane.

Answer

The derivation of $a^{b+ic}$ comes from Euler's Identity: $$a^{b+ic}=a^ba^{ic}=a^b e^{\ln(a)ic}=a^b(\cos(c\ln a)+i\sin(c\ln a))$$

The geometric interpretation is that of a rotation of the vector $\langle a^b, 0\rangle$ by an angle of $c\ln a$.