calculus - Prove the following inequality using the Mean Value Theorem

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.