Exponent Combination Laws/Negative Power

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $a \in \R_{>0}$ be a strictly positive real number.

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

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


Then:

$a^{-x} = \dfrac 1 {a^x}$


Proof

\(\ds a^{-x}\) \(=\) \(\ds \map \exp {-x \ln a}\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds \paren {\map \exp {x \ln a} }^{-1}\) Exponential of Product
\(\ds \) \(=\) \(\ds \frac 1 {\map \exp {x \ln a} }\)
\(\ds \) \(=\) \(\ds \frac 1 {a^x}\) Definition of Power to Real Number

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Exponent_Combination_Laws/Negative_Power&oldid=676901"