Determining whether equation $\ln x=\frac{1}{3}x$ has at least 3 solutions I'm trying to say whether
$$\ln x=\frac{1}{3}x$$
Has at least 2 solutions, and after that the problem asks us to tell if the third solution exists...
So I managed to find 1 solution: $f(x)=\ln x - \frac{1}{3} x$, $f(e)>0, f(1)=-\frac{1}{3} < 0$, so by the mean value theorem we have $f(y)=0$ for some $y\in [1,e]$.
What to do about the other two?
UPDATE: I haven't covered derivatives yet in my course...

If you look at the graph of the given equation, the plot of ln(x) bends towards X-axis after the second point of intersection while the plot of $\frac13x$ continues straight above. So, there won't be a third solution of the equation.
