Power Series Expansion for General Exponential Function

Theorem
Let $a \in \R_{> 0}$ be a (strictly) positive real number.

Then:

Then:

Proof
By definition of a power to a real number:
 * $a^x = \map \exp {x \ln a}$

As $x \ln a$ is itself a real number, we can use Power Series Expansion for Exponential Function:

substituting $x \ln a$ for $x$.