Mean Value theorem problem?(inequality)
Apply the mean value theorem to the function $x\mapsto \ln x$ on the interval $[a,b]$: $\exists c\in(a,b)$ such that $$1-\frac b a=\frac {b-a} b<\ln b-\ln a=\ln \frac{b} {a}=(b-a)\frac 1 c<\frac {b-a} a=\frac b a-1$$
HINT : $$1-\frac ab\lt \ln b-\ln a\lt \frac ba -1$$ $$\iff \frac{1-\frac ab}{b-a}\lt\frac{\ln b-\ln a}{b-a}\lt\frac{\frac ba -1}{b-a}$$ $$\iff \frac{b-a}{b(b-a)}\lt \frac{\ln b-\ln a}{b-a}\lt\frac{b-a}{a(b-a)}$$