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