Question

For what values of a does the equation | ln x| - ax = 0 possess two roots?

Answer 1

$\left|lnx\right|-ax=0$

$y=\left|lnx\right|\&y=ax$

$y^1=\frac{1}{x}$

$y-y_0=y^1(x-x_0)$

$\Rightarrow y-{lnx}_0=\frac{1}{x_0}(x-x_0)\Rightarrow y=\frac{x}{x_0}+({lnx}_0-1)$

On comparing, $lnx=1$ $y_0={ax}_0$

$\Rightarrow x_0=e$ $\Rightarrow a=\frac{1}{e}$