Prove the following inequality using the Mean Value Theorem

Question

I want to show that

$$\log(2+x) - \log(x) \lt \frac{2}{x}$$ $\forall x \in \mathbb{R^+}$

I know I need to apply the Mean Value Theorem to find an upper bound of the function to the left and show that it is smaller than $\frac{2}{x}$, but I can't find the correct upper bound. I've tried multiple variations of the inequality. My teacher also said that I only needed to check in the interval from 0 to 2, but I'm not sure why.

$\exists c\in(x,x+2)\Rightarrow (log)'(c)=\frac{log(2+x)-log(x)}{2+x-x}$ — Mateus Rocha, Jul 11 '18 at 00:41
@MateusRocha why are you applying to $log(x)$? It should be to $log(2+x)-log(x)$ — chilliefiber, Jul 11 '18 at 00:45
using what I've commented, $\frac{1}{c}=\frac{log(2+x)-log(x)}{2}\Rightarrow \frac{2}{c}=log(2+x)-log(x)$. But $c>x$, so $\frac{2}{c}<\frac{2}{x}$ — Mateus Rocha, Jul 11 '18 at 00:48
Essentially the same problem as in https://math.stackexchange.com/questions/2067473/show-that-logx1-logx-frac1x-for-x-0. — Martin R, Jul 11 '18 at 07:48

score 1 · Answer 1 · answered Jul 11 '18 at 00:52

1

For every real $\alpha>0$, let $f(x)=\ln(x+\alpha)$. Now with Mean Value Theorem in $[0,2]$ : $$f(2)-f(0)=f'(c)(2-0)$$ for a $c\in(0,2)$. This gives us $\ln(2+\alpha)-\ln\alpha<\dfrac{2}{\alpha}$.

answered Jul 11 '18 at 00:52

Nosrati

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Prove the following inequality using the Mean Value Theorem

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