Given two positive real numbers http://latex.codecogs.com/gif.latex?a,b (a<b), prove that

$\displaystyle \sqrt{ab}<[(a-b)/(\ln(a)-\ln(b))]<(a+b)/2$

One thing that strikes me first is that the middle term is actually the point in (a,b) which satisfies Lagrange's Mean Value Theorem for the function $\displaystyle \ln(x)$

But I can't think of any way of proving that the point lies between the geometric mean and the arithmetic mean of the numbers a,b.

Any suggestions??