Question

If $0<a<b,$ then which of the following is correct :

• A. $\frac{(b-a)}{a}<\ln \frac{b}{a}<\frac{(b-a)}{b}$
• B. $\frac{(b-a)}{b}<\ln \frac{b}{a}<\frac{(a-b)}{b}$
• C. $\frac{(b-a)}{b}<\ln \frac{b}{a}<\frac{(b-a)}{a}$
• D. $\frac{(b-a)}{a}<\ln \frac{b}{a}<\frac{(a-b)}{a}$

Answer 1

The correct option is C $\frac{(b-a)}{b}<\ln \frac{b}{a}<\frac{(b-a)}{a}$

Let $f\left(x\right)=\ln x$
Applying lagrange formula to it on the interval $[a,b]$
$\Rightarrow \frac{f\left(b\right)-f\left(a\right)}{b-a}=f^{\prime }\left(c\right),c\in (a,b)$
Hence, $\frac{\ln b-\ln a}{b-a}=\frac{1}{c}$
$\Rightarrow \ln \frac{b}{a}=(b-a)\frac{1}{c}$
The right hand side of this equation is minimum when $c=b$ and takes a maximum at $c=a$
$\therefore \frac{b-a}{b}<\ln \frac{b}{a}<\frac{b-a}{a}$