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 |
$\blacksquare$