# What is the equation of the line normal to  f(x)=lnx-x at  x=2?

Jun 9, 2016

$y = {\log}_{2} \left(2\right) - 2 + 2 \left(x - 2\right)$

#### Explanation:

taking $f \left(x\right) = L o {g}_{e} \left(x\right) - x$ the tangent space at each point is obtained
with

$m \left(x\right) = \frac{\mathrm{df}}{\mathrm{dx}} = \frac{1}{x} - 1$

Also the normal space direction is obtained with

$n \left(x\right) = - \frac{1}{m \left(x\right)}$

In this case, at point ${p}_{0} = \left\{{x}_{0} , {y}_{0}\right\} = \left(2 , {\log}_{e} \left(2\right) - 2\right\}$ we have:

${n}_{0} \left(2\right) = - \frac{1}{\left(- \frac{1}{2}\right)} = 2$

Finally the normal straight at point ${p}_{0}$ reads

$y = {\log}_{2} \left(2\right) - 2 + 2 \left(x - 2\right)$