I'm not really sure how to go about this problem, as I've never encountered anything similar before. I'm supposed to find all the values $m$ for which the following equation has $3$ distinct real solutions: $$|\ln(x)| = mx$$
This is taken from my $12$th grade calculus course, but no indications are shown.
Any ideas would be welcome !
EDIT: By plotting, I found that $m \in (0,\frac{1}{e})$ . But I'm much more interested in how an analytical solution would look like.
Let $\Gamma$ be the graph of $y=\ln x$. The ligne passing by $(0,0)$ and tangent to $\Gamma$ has the equation $y=\frac{x}{e}$, the point of tangency being $(e,1)$.
For all $m>\frac1e$ the ligne $y=mx$ does not touch $\Gamma$ and for $m\le0$ there is just a point.
It follows that for $0<m<\frac1e$ the ligne $y=mx$ cuts $\Gamma$ in two point so the ligne $y=-mx$ cuts $\Gamma$ in just a point.
Hence, since $|\ln x|=mx\iff\ln x=mx\space \text{and}\space \ln x=-mx$, the asked values are such that $$0<m<\frac 1e$$