Logarithm of Power/General Logarithm

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

Then:

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

Proof

Let $y = r \log_a x$.

Then:

 $\ds a^y$ $=$ $\ds a^{r \log_a x}$ $\ds$ $=$ $\ds \paren {a^{\log_a x} }^r$ Exponent Combination Laws $\ds$ $=$ $\ds x^r$ Definition of Logarithm base $a$

The result follows by taking logs base $a$ of both sides.

$\blacksquare$

Sources

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