Logarithm of Power/Natural Logarithm/Integer Power

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

Let $n \in \R$ be any integer.

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

Then:
 * $\map \ln {x^n} = n \ln x$

Proof
From Logarithm of Power/Natural Logarithm/Natural Power, the theorem is already proven for positive integers.

Let $j \in \Z_{<0}$.

Let $-j = k \in Z_{>0}$.

Then: