We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term W := 1(-infinity,0](center dot)exp(center dot) * rho satisfy an Oleinik-type entropy condition. More precisely, under different sets of assumptions on the velocity function V, we prove that W satisfies a one-sided Lipschitz condition and that V '(W)W partial derivative xW satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.