Exponent Combination Laws/Power of Product

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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:

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

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

$\blacksquare$

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