Power Function on Base Greater than One is Strictly Increasing/Rational Number
Jump to navigation
Jump to search
Theorem
Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Q \to \R$ be the real-valued function defined as:
- $\map f q = a^q$
where $a^q$ denotes $a$ to the power of $q$.
Then $f$ is strictly increasing.
Proof
Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z$ are integers and $s, u \in \Z_{>0}$ are strictly positive integers.
Let $\dfrac r s < \dfrac t u$.
From Ordering of Reciprocals:
- $0 < \dfrac 1 a < 1$
So:
| \(\ds \paren {\frac 1 a}^{t / u}\) | \(<\) | \(\ds \paren {\frac 1 a}^{r / s}\) | Power Function on Base between Zero and One is Strictly Decreasing: Rational Number | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sqrt [u] {\paren {\frac 1 a}^t}\) | \(<\) | \(\ds \sqrt [s] {\paren {\frac 1 a}^r}\) | Definition of Rational Power | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sqrt [u] {\paren {\frac 1 {a^t} } }\) | \(<\) | \(\ds \sqrt [s] {\paren {\frac 1 {a^r} } }\) | Real Number to Negative Power: Integer | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 {\sqrt [u] {a^t} }\) | \(<\) | \(\ds \frac 1 {\sqrt [s] {a^r} }\) | Root of Reciprocal is Reciprocal of Root | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 {a^{t / u} }\) | \(<\) | \(\ds \frac 1 {a^{r / s} }\) | Definition of Rational Power | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a^{r / s}\) | \(<\) | \(\ds a^{t / u}\) | Ordering of Reciprocals |
Hence the result.
$\blacksquare$