Exponent Combination Laws/Power of Product

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Theorem

Let $a, b \in \R_{\ge 0}$ 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:

$\paren {a b}^x = a^x b^x$


Proof

\(\ds \paren {a b}^x\) \(=\) \(\ds \map \exp {x \map \ln {a b} }\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds \map \exp {x \ln a + x \ln b}\) Sum of Logarithms
\(\ds \) \(=\) \(\ds \map \exp {x \ln a} \map \exp {x \ln b}\) Exponential of Sum
\(\ds \) \(=\) \(\ds a^x b^x\) Definition of Power to Real Number

$\blacksquare$


Sources

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