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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?

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Yes this is correct.

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

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