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