Logarithm of Power/Natural Logarithm/Proof 2

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
By hypothesis, $\ln a = b$.

Multiplying both sides by $c$:


 * $c \ln a = c b$

But we proved above that:


 * $c b = \ln a^c$