Logarithmic Sobolev inequalities are a fundamental class of inequalities that play an important role in information theory. They play a key role in establishing concentration inequalities and in obtaining quantitative estimates on the convergence to equilibrium of Markov processes. More recently, deep links have been established between logarithmic Sobolev inequalities and strong data processing inequalities. In this paper we study logarithmic Sobolev inequalities from a computational point of view. We describe a hierarchy of semidefinite programming relaxations which give certified lower bounds on the logarithmic Sobolev constant of a finite Markov operator, and we prove that the optimal values of these semidefinite programs converge to the logarithmic Sobolev constant. Numerical experiments show that these relaxations are often very close to the true constant even for low levels of the hierarchy. Finally, we exploit our relaxation to obtain a sum-of-squares proof that the logarithmic Sobolev constant is equal to half the Poincaré constant for the specific case of a simple random walk on the odd $n$-cycle, with $n\in\{5,7,\dots,21\}$. Previously this was known only for $n=5$ and even $n$.