How to prove the logaritmic inequality: $\ 1- \frac{a}{b} < \ln(b)-\ln(a)<\frac{b}{a} - 1$?

Question

I'm stuck in this exercise:

Prove the following inequality: $$\ 1- \frac{a}{b} < \ln(b)-\ln(a)<\frac{b}{a} - 1$$

Answer 1

Hint: by the mean value theorem, we have $\frac{\ln b-\ln a}{b-a}=\frac{1}{c}$ for some $c\in (a,b)$. To conclude, note that $\frac{1}{b}<\frac{1}{c}<\frac{1}{a}$.

Answer 2

Hint:

If $b>a$, $$\ln b-\ln a=\ln(b/a)=\int_1^{b/a}\frac{dt}t<\int_1^{b/a}dt=\frac ba-1$$

Answer 3

I think, $a$ and $b$ should be positives, otherwise, your inequality is wrong.

Also, it should be not strong inequality, otherwise, it's wrong for $a=b$.

Let $\frac{b}{a}=x$.

Thus, we need to prove that $$1-\frac{1}{x}\leq\ln(1+x-1)\leq x-1.$$

I think now you can end it.