TITLE: Prove that if $a>b>0$, then $\frac{a-b}{a}<\ln a - \ln b <\frac{a-b}{b}$.
QUESTION [5 upvotes]: Prove that if $a>b>0$, then 
$$
\frac{a-b}{a}<\ln a - \ln b <\frac{a-b}{b}.
$$
This might/should be an application of the Mean Value Theorem

REPLY [2 votes]: $$\frac{a-b}{b}<\log a - \log b <\frac{a-b}{b}\iff\frac{1}{b}(a-b)<\log a-\log b<\frac{1}{a}(a-b)\iff$$
$$\frac{1}{b}(a-b)<\int\limits_b^a\frac{dx}{x}<\frac{1}{a}(a-b)$$
The above is just an application Integral MVT (formula $(2)$) for the descending function $\,\frac{1}{x}\,$