I suppose the method would be to derive ln(xy) = ln(x) + ln(y) and the corresponding exp(x + y) = exp(x)exp(y), then see how this lets exp(y ln(x)) coincide wit

I suppose the method would be to derive ln(xy) = ln(x) + ln(y) and the corresponding exp(x + y) = exp(x)exp(y), then see how this lets exp(y ln(x)) coincide with repeated multiplication for integer y. Connecting this to the series definition of exp(x) would also take some work, but my textbook wasn't very big on series definitions in general.