Roots and complex numbers

cjellison
There are n nth roots to every complex number (except zero).

My question: How many "roots" are there when you take a complex number to an irrational or transcendental number. For that matter, how do we define raising a number to an irrational number? How do we define raising a number to a transcendental number?

Hmm...how is this defined? By a taylor series?

$$e^{1/e}$$

$$z^w := \exp(w \mathop{\mathrm{Log}} z)$$