For which values of $m$, $f(x)=mx$ intersect the function $g(x)=\log x$?

Question

For which values of $m$, the function, $f(x)=mx$ intersect the function, $g(x)=\log x$

I suppose that this problems reduce to the next form.

Find for which values of m, exist solution for the equation, $\quad mx=\log x$.

This could be a good aproach if I would have good tools for solving this equations, but this is not the case, So a second choice is to have clues supposing thing from the graph of log x, avoiding formality.

Any deeper and accurate answer?

Answer 1

There will be some $m_0$ for which $m_0x$ intersects $\log x$ exactly once. So the answer is, the set of $m\le m_0$ are the $m$ for which $mx$ crosses $\log x$ (this comes from looking at a picture). To find $m_0$, find the equation for the tangent line of $\log x$ at each point $x_0$, then see which tangent line has a $y$-intercept of $0$.

Answer 2

Hint: For $m\le 0$, it is clear (why?) that an intersectoin occurs. For $m>0$ first answer the question: For which value of $m$ is the line $y=mx$ a tangent line to the graph of $g$?