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:

\(\ds 0\)

\(=\)

\(\ds \ln 1\)

Logarithm of 1 is 0

\(\ds \)

\(=\)

\(\ds \map \ln {x^k x^{-k} }\)

\(\ds \)

\(=\)

\(\ds \map \ln {x^k} + \map \ln {x^{-k} }\)

\(\ds \)

\(=\)

\(\ds k \ln x + \map \ln {x^{-k} }\)

Logarithm of Power/Natural Logarithm/Natural Power

\(\ds \leadsto \ \ \)

\(\ds \map \ln {x^{-k} }\)

\(=\)

\(\ds -k \ln x\)

\(\ds \leadsto \ \ \)

\(\ds \map \ln {x^j}\)

\(=\)

\(\ds j \ln x\)

$\blacksquare$