Math Help - Lagrange's Theorem(?)

1. Lagrange's Theorem(?)

Given two positive real numbers $a,b (a, prove that

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

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 $\ln(x)$

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??

2. maybe you can consider this

let $f(x)=\frac{x-a}{lnx-lna}-\sqrt{ax}$, then see $f'(x)$ and $f''(x)$.

another possible way: through Taylor's Mean Value Theorem. But I have not tried it.