A short paper, whose main point is to bring together ideas in the literature on bounds for total variation and relate them to relative entropy. A new inequality is derived for the relative entropy of a Markov chain. which yields a very short proof of convergence of countable Markov chains in relative entropy. It also enables bounds using the second largest eigenvalue of the chain (or an associated chain in the non-reversible case) and Fourier bounds for random walks on groups. These are based on total variation bounds already available, and shows that for many random walks, rates of convergence in relative entropy will be the same as in total variation. However, we also mention an example in which rates of convergence differ.