Mathematical question
So, it's possible to expand a function e^x into a series of x^n.
How can I do the reverse?
How could I expand x^n into a series of exponential functions m^x ?
Google always brings me back to a Taylor-expansion of e^x.
Is there some special name for the expansion I'm looking for?
Where should I start looking?
= new reply since forum marked as read
Are you familiar with Fourier series? Fourier series expand a function very similarly to how you describe:
f(x) = sum(c_n e^(2*pi*i(n/T) x)), with T the period.
Due to the presence of the imaginary unit "i" this represents f in terms of frequencies, so it is not precisely as you ask.
The generalization of this is the Fourier transform which is a representation for non-periodic functions (look up the Gibbs phenomena for how Fourier series fail).
To your direct question, there is a reason you are having a difficult time: there is no compelling reason for such an expansion. A good reason that people talk about Taylor series is that the polynomial representation (powers of x) allows you to obtain estimates for the error you commit when you only use a limited number of terms (the full Taylor series is infinitely long). Exponential functions grow very, very, very, fast so the error estimates would be a nightmare.
This does not apply to the Fourier series I listed above since exponentials of imaginary units can be bounded by 1. 
I hope this helps.
