For $a\gt b\gt 0$, prove that $\dfrac{a+b}{2}\gt \dfrac{a-b}{\ln a-\ln b}$.

For $a\gt b\gt 0$, prove that $\dfrac{a+b}{2}\gt \dfrac{a-b}{\ln a-\ln b}$.

There sure is many way to prove this problem, but I am going to show you one graphical method that if we recognize some function is always greater than the other in certain interval, then it makes the problem all that easier for us to crack.