Derivative of General Logarithm Function

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $a \in \R_{>0}$ such that $a \ne 1$

Let $\log_a x$ be the logarithm function to base $a$.

Then:

$\map {\dfrac \d {\d x} } {\log_a x} = \dfrac {\log_a e} x$


Proof

From Change of Base of Logarithm, we have:

$\log_e x = \dfrac {\log_a x} {\log_a e}$

from which:

$\log_a x = \log_a e \log_e x$

In this context, $\log_a e$ is a constant.

The result then follows from Derivative of Natural Logarithm Function and Derivative of Constant Multiple.

$\blacksquare$