Logarithm of Power/General Logarithm

From ProofWiki
Jump to navigation Jump to search

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

WARNING: This link is broken. Amend the page to use {{KhanAcademySecure}} and check that it links to the appropriate page.