The reason to write C e^(x ln a) is because when you go to differentiate it or integrate it, you don't need to remember a formula for for exponential expressions with other bases.

d/dx (e^(x ln a)) = ln a e^(x ln a) = ln a a^x

So you can either turn a^x into e^(x ln a) or you can remember d/dx (a^x) = ln a a^x as a separate "rule" of differentiation.

If it were me, I'd err on the side of memorizing fewer things.