# How do you differentiate log_2 (x)?

Feb 8, 2016

$\frac{d}{\mathrm{dx}} {\log}_{2} \left(x\right) = \frac{1}{x \cdot \ln \left(2\right)}$

#### Explanation:

This follows from the general formula:
$\textcolor{w h i t e}{\text{XXX}} \frac{d}{\mathrm{dx}} \left({\log}_{a} \left(x\right)\right) = \frac{1}{x \cdot \ln \left(a\right)}$

Feb 8, 2016

"d"/("d"x) [log_2(x)] = 1/(xln(2))

#### Explanation:

As we know how to differentiate $\ln \left(x\right)$, we should change the base of the logarithm first.

The according formula to change a logarithmic expression from the base $a$ to the base $b$ is

${\log}_{\textcolor{red}{a}} \left(\textcolor{b l u e}{x}\right) = {\log}_{b} \frac{\textcolor{b l u e}{x}}{\log} _ b \left(\textcolor{red}{a}\right)$

You can apply the formula as follows:

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

As $\frac{1}{\ln} \left(2\right)$ is just a constant and the derivative of $\ln \left(x\right)$ is $\frac{1}{x}$, our derivative is:

"d"/("d"x) [log_2(x)] = "d"/("d"x) [ln(x) / ln(2) ] = 1/ln(2) * 1/x = 1/(xln(2))