Exponent Combination Laws/Negative Power

Theorem

Let $x \in \R$ be a real number.

Let $a^x$ be defined as $a$ to the power of $x$.

Then:

$a^{-x} = \dfrac 1 {a^x}$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a^{-x}$	$=$	$\displaystyle $	$\displaystyle \exp \left({-x \ln a}\right)$	$\displaystyle $	$\displaystyle $	Definition of Power to Real Number
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle \left({\exp \left({x \ln a}\right)}\right)^{-1}$	$\displaystyle $	$\displaystyle $	Exponent of Product
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle \frac 1 {\exp \left({x \ln a}\right)}$	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle \frac 1 {a^x}$	$\displaystyle $	$\displaystyle $	Definition of Power to Real Number

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a^{-x}\)	\(=\)	\(\displaystyle \)	\(\displaystyle \exp \left({-x \ln a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	Definition of Power to Real Number
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \left({\exp \left({x \ln a}\right)}\right)^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	Exponent of Product
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \frac 1 {\exp \left({x \ln a}\right)}\)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \frac 1 {a^x}\)	\(\displaystyle \)	\(\displaystyle \)	Definition of Power to Real Number