Exponent Combination Laws/Power of Product

From ProofWiki
Jump to: navigation, search

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:

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


Proof

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

$\blacksquare$


Sources