calculus - Mean Value theorem problem?(inequality)

I'm trying to solve this mean value theorem problem but confused where to start,

If $0

Can someone please lend me a hand?

Answer

Apply the mean value theorem to the function $x\mapsto \ln x$ on the interval $[a,b]$: $\exists c\in(a,b)$ such that

$$1-\frac b a=\frac {b-a} b<\ln b-\ln a=\ln \frac{b} {a}=(b-a)\frac 1 c<\frac {b-a} a=\frac b a-1$$