I'm trying to proof the following inequality: 
$x+1<e^{x}<2x+1$ for $0<x\leq log2$   

I used the MVT on with $f(t)=e^{t}$ on the interval $(0,x)$ to proof that $x+1<e^{x}$ and I tried to do the same with $(x, log2)$ but I only got to $e^{x}<2x-2log2+2$.   
 Any help or hint will be appreciated.