Derivative of Logarithm Functions

$$ 1. \ \frac {d}{dx} \ln x = \frac 1x $$

$$ 2. \ \frac {d}{dx} \log_{a} {x} = \frac {1}{x \ln a} $$

$$ 3. \ \frac {d}{dx} \ln f(x) = \frac {1}{f(x)} \frac {d}{dx} f(x) $$

$$ 4. \ \frac {d}{dx} \log_{a} {f(x)} = \frac {1}{f(x) \ln a} \frac {d}{dx} f(x) $$

Example:

$$ Differentiate \ g(t)= 4 \log_{3} {t} - \ln t $$

Solution:

$$ g'(t)= \frac {4}{t \ln (3)} - \frac 1t $$