Derivative of General Logarithm Function/Proof

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

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Log to base $a$