Logarithm of Power/Natural Logarithm/Proof 3

Theorem
Let $x \in \R$ be a strictly positive real number.

Let $a \in \R$ be a real number such that $a > 1$.

Let $r \in \R$ be any real number.

Let $\ln x$ be the natural logarithm of $x$.

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

Proof
Here we adopt the definition of $\ln x$ be:
 * $\displaystyle \ln x := \int_1^x \dfrac {\mathrm d t} t$