# Power Series Expansion for General Exponential Function

## Theorem

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

Then:

Then:

 $\ds \forall x \in \R: \ \$ $\ds a^x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {\left({x \ln a}\right)^n} {n!}$ $\ds$ $=$ $\ds 1 + x \ln a + \frac {\left({x \ln a}\right)^2} {2!} + \frac {\left({x \ln a}\right)^3} {3!} + \cdots$

## Proof

By definition of a power to a real number:

$a^x = \exp \left({x \ln a}\right)$

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

 $\ds \forall x \in \R: \ \$ $\ds \exp x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ $\ds$ $=$ $\ds 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots$

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

$\blacksquare$