# Log rule in calculus

## Statement

The derivative of a logarithmic function is given by:

$$\tfrac{d}{dx}\bigg(\log_b(x)\bigg) = \frac{1}{x * \ln(b)}$$

## Proof

Define function $f$.

$$f(x) = \log_b(x)$$

Rewrite the logarithm using the change of base formula with base $e$ and write the denominator as a coefficient.

$$f(x) = \frac{\ln(x)}{\ln(b)} = \frac{1}{\ln(b)} * \ln(x)$$

Differentiate this function by using the coefficient rule and the derivative of $\ln(x)$.

$$f'(x) = \frac{1}{\ln(b)} * \frac{1}{x}$$

Finally, multiply the fractions.

$$f'(x) = \frac{1}{x * \ln(b)}$$