Differentiation of Logarithmic Functions

Table of Contents

For any base $a > 0 (a \neq 1)$ and $x > 0$, the derivative of the logarithmic function is given by the following formula:

$$\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$$

where $a$ is the base of the logarithm and $\ln(a)$ is the natural logarithm of $a$.

The derivative of the natural logarithm function is:

$$\frac{d}{dx}[\ln(x)] = \frac{1}{x}$$

where $\ln(x) = \log_e(x)$, with $e$ being Euler’s number (approximately 2.71828), and $\ln(e) = 1$.

The derivative of the common logarithm (base 10) function is:

$$\frac{d}{dx}[\log_{10}(x)] = \frac{1}{x \ln(10)}$$