Derivative of log x

The natural logarithm $\log x$ differentiates to $\frac{1}{x}$. To include negative $x$, write $\log|x|$; its derivative is the same $\frac{1}{x}$.

For a base $a$ other than $e$, divide by the constant $\ln a$: the derivative is $\frac{1}{x\ln a}$.

Examples

1. 01Differentiate$\displaystyle \frac{d}{dx} \log_e x$AnswerHide

   Answer$\displaystyle \frac{1}{x}$
2. 02Differentiate$\displaystyle \frac{d}{dx} \log 2x$AnswerHide

   Answer$\displaystyle \frac{1}{x}$
3. 03Differentiate$\displaystyle \frac{d}{dx} \log(x^{2}+1)$AnswerHide

   Answer$\displaystyle \frac{2x}{x^{2}+1}$
4. 04Differentiate$\displaystyle \frac{d}{dx}\, x\log x$AnswerHide

   Answer$\displaystyle \log x + 1$
5. 05Differentiate$\displaystyle \frac{d}{dx} (\log x)^{2}$AnswerHide

   Answer$\displaystyle \frac{2\log x}{x}$
6. 06Differentiate$\displaystyle \frac{d}{dx} \log_{2} x$AnswerHide

   Answer$\displaystyle \frac{1}{x\log 2}$

FAQ

Q1What is the derivative of a base-10 logarithm?Show answerHide

A

$(\log_a x)'=\frac{1}{x\log a}$. Only when the base is e does it become $\frac{1}{x}$.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05