Mean Value theorem problem?(inequality)

Question

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

If $0<a<b$ prove that $(1-\frac{a}{b})<\ln\frac{b}{a}<\frac{b}{a}-1$

Can someone please lend me a hand?

See also: Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$, Showing $\frac{x}{1+x}<\log(1+x)<x$ for all $x>0$ using the mean value theorem. (And other posts linked there.) — Martin Sleziak, Jun 13 '20 at 09:26

score 5 · Accepted Answer · answered Dec 26 '13 at 06:58

Apply the mean value theorem to the function $x\mapsto \ln x$ on the interval $[a,b]$: $\exists c\in(a,b)$ such that $$1-\frac b a=\frac {b-a} b<\ln b-\ln a=\ln \frac{b} {a}=(b-a)\frac 1 c<\frac {b-a} a=\frac b a-1$$

score 2 · Answer 2 · answered Dec 26 '13 at 06:53

2

HINT : $$1-\frac ab\lt \ln b-\ln a\lt \frac ba -1$$ $$\iff \frac{1-\frac ab}{b-a}\lt\frac{\ln b-\ln a}{b-a}\lt\frac{\frac ba -1}{b-a}$$ $$\iff \frac{b-a}{b(b-a)}\lt \frac{\ln b-\ln a}{b-a}\lt\frac{b-a}{a(b-a)}$$

answered Dec 26 '13 at 06:53

mathlove

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Mean Value theorem problem?(inequality)

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