Logarithm of Power/Natural Logarithm/Rational Power

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 $t$

$\blacksquare$