Is $f(x)$ convex if $\log f(x)$ is convex? One of the convex composition rules states that $h(g(x))$ is convex if $h(x)$ is convex and non-decreasing, and $g(x)$ is convex.
Now I want to go the other way - I know that $\log(f(x))$ is convex and want to learn about $f(x)$. Let's assume $f(x)$ is always positive.
Now I reason that we can use this composition rule to show convexity of $f(x)$. Since $e^x$ is convex and non-decreasing, and we know $\log f(x)$ is convex, then their composition $e^{\log(f(x))}=f(x)$ is convex.
Is this proof correct?

Yes this is correct.
By the way, $f$ is called a log-convex function. 
