Exponent Combination Laws/Power of Quotient

Theorem

Let $a, b \in \R_+$ be positive real numbers.

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

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

Then:

$\left({\dfrac a b}\right)^x = \dfrac{a^x}{b^x}$

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

$\blacksquare$

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