If $f$ is log-convex then $f$ is convex

Here's my attempt:$f$ is log-convex. Then:

$\log f(\lambda x + (1-\lambda)y )\leq \lambda \log f(x) + (1-\lambda) \log f(y)$

As $e^x$ is increasing, we can apply it to the inequation without changing its signal. Then:

$f(\lambda x + (1-\lambda)y )\leq f(x)^{\lambda} . f(y)^{(1-\lambda) }$

The left side is alright, but I can't see how to conclude that $f$ is convex.Can someone help me?

Thanks.