Derivative of General Logarithm Function

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

 $\ds \map {\dfrac \d {\d x} } {\log_a x}$ $=$ $\ds \map {\dfrac \d {\d x} } {\dfrac {\log_e x} {\log_e a} }$ Change of Base of Logarithm $\ds$ $=$ $\ds \dfrac 1 {\log_e a} \map {\dfrac \d {\d x} } {\log_e x}$ Derivative of Constant Multiple $\ds$ $=$ $\ds \dfrac 1 {\log_e a} \dfrac 1 x$ Derivative of Natural Logarithm $\ds$ $=$ $\ds \dfrac {\log_a e} x$ Logarithm of Reciprocal

$\blacksquare$

Also presented as

This result can also be seen presented as:

$\map {\dfrac \d {\d x} } {\log_a x} = \dfrac 1 {x \ln a}$

where $\ln a := \log_e a$ is the natural logarithm of $a$.

This is seen to be equivalent to the given form by Logarithm of Reciprocal:

$\dfrac 1 {\ln a} = \log_a e$