# What is the equation of the normal line of f(x)=ln2x-x at x=4 ?

Dec 6, 2016

$y = \frac{4}{3} x + 3 \ln 2 - \frac{28}{3}$

#### Explanation:

The equation of the line normal to the curve $y = f \left(x\right)$ in the point $x = \overline{x}$ is given by:

$y = f \left(\overline{x}\right) - \frac{1}{f ' \left(\overline{x}\right)} \left(x - \overline{x}\right)$

Given $\overline{x} = 4$:

$f \left(x\right) = \ln 2 x - x$
$f \left(4\right) = \ln 8 - 4 = 3 \ln 2 - 4$

$f ' \left(x\right) = \frac{1}{x} - 1$
$f ' \left(4\right) = - \frac{3}{4}$

The equation of the normal line is:

$y = \frac{4}{3} \left(x - 4\right) + 3 \ln 2 - 4 = \frac{4}{3} x + 3 \ln 2 - \frac{28}{3}$