How to prove a nonlinear tracendent equation has two positive roots?

Question

How to show (but do not use numerical software such as Mathematica, Matlab...etc.) that this equation \begin{equation} \frac{u (83811 u-88223)+18076}{396-3276 u}-\frac{10 \log (u)}{3}-\frac{1}{2} \log \left(\frac{98 u+14}{273 u-33}\right)=0 \end{equation} has two roots $u_1$ and $u_2$ with $u_1,u_2>0$?

Answer 1

Well, you can find the answers numerically first (but do not include in your write up). Make sure the function above satisfies intermediate-value theorem. And then just show there is a sign change around the values near the numerically determined solutions, and therefore there must be two solutions. Something like that should work.