Power Series Expansion for General Exponential Function

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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 $x$.

$\blacksquare$


Sources