Exponents And Irrationals

The meaning of b^a is clear when a and b are rational. It is even understandable, in terms of limits, when one of them is irrational. What about when a and b are both irrational? For all reals a and b, exp(a·ln(b)) is a useful definition. ln(x) is the area under the plot of 1 / t from t = 1 to t = x for x > 0. exp(x) is the inverse of ln(x).