Logarithm of Power/Natural Logarithm/Rational Power
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Theorem
Let $x \in \R$ be a strictly positive real number.
Let $r \in \R$ be any rational number.
Let $\ln x$ be the natural logarithm of $x$.
Then:
- $\map \ln {x^r} = r \ln x$
Proof
Let $r = \dfrac s t$, where $s \in \Z$ and $t \in \Z_{>0}$.
First:
\(\ds \map \ln x\) | \(=\) | \(\ds \map \ln {x^{t / t} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\paren {x^{1 / t} }^t}\) | Product of Indices of Real Number/Rational Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds t \map \ln {x^{1 / t} }\) | Logarithm of Power/Natural Logarithm/Integer Power | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \map \ln {x^{1 / t} }\) | \(=\) | \(\ds \frac 1 t \map \ln x\) | dividing both sides by $t$ |
Thus:
\(\ds \map \ln {x^{s / t} }\) | \(=\) | \(\ds \map \ln {\paren {x^{1 / t} }^s}\) | Product of Indices of Real Number/Rational Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds s \map \ln {x^{1 / t} }\) | Logarithm of Power/Natural Logarithm/Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac s t \map \ln x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds r \map \ln x\) | Definition of $s$ and $t$ |
$\blacksquare$