Logarithm of Power

Theorem

Natural Logarithms

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:

$\map \ln {x^r} = r \ln x$

General Logarithms

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 $\log_a x$ be the logarithm to the base $a$ of $x$.

Then:

$\map {\log_a} {x^r} = r \log_a x$

Sources

• For a video presentation of the contents of this page, visit the Khan Academy.