We give a sufficient and necessary condition for a probability measure $μ$ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto $μ$. The main tool in the proof is the theory of weak transport costs. As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.