Exponent Combination Laws/Power of Quotient

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Theorem

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers.

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

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


Then:

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


Proof

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

$\blacksquare$