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:
 * $\ln \left({ x^r }\right) = r \ln x$

Proof
Let $r = \dfrac{s}{t}$, where $s \in \Z$ and $t \in \Z_{>0}$

First:

Thus: