We show that for any $α>0$ the Rényi entropy of order $α$ is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The first case occurs for $α\in (0,α^*]$ and the second case for $α\in [α^*,\infty)$, where $α^*$ satisfies the equation $\frac{1}{α^*-1}\log α^*= \frac12 \log 6$, that is $α^* \approx 1.241$. Using those results, we prove that one-sided exponential distribution minimizes Rényi entropy of order $α\geq 2$ among all log-concave random variables with fixed variance.