Differentiating Logarithmic Functions without Base e

Key Questions

• By Change of Base Formula: ${\log}_{b} x = \frac{{\log}_{a} x}{{\log}_{a} b}$,

$y = {\log}_{3} x = \frac{\ln x}{\ln 3}$

By taking the derivative,

$y ' = \frac{\frac{1}{x}}{\ln 3} = \frac{1}{\left(\ln 3\right) x}$

$- \tan \frac{x}{\ln} \left(2\right)$

Explanation:

$f \left(x\right) = {\log}_{2} \left(\cos \left(x\right)\right) = \ln \frac{\cos \left(x\right)}{\ln} \left(2\right)$

$\frac{1}{\ln} \left(2\right)$ is just a constant and can be ignored.

$\left(\ln \left(u\right)\right) ' = \frac{u '}{u}$

$u = \cos \left(x\right) , u ' = - \sin \left(x\right)$

$f ' \left(x\right) = \frac{1}{\ln} \left(2\right) \cdot \frac{- \sin \left(x\right)}{\cos} \left(x\right) = - \tan \frac{x}{\ln} \left(2\right)$

• Change of Base Formula

${\log}_{b} x = \frac{{\log}_{a} x}{{\log}_{a} b}$,

where $a$ is any positive number except $1$.

I hope that this was helpful.