A sequence ( sn) of real numbers is said to be summable to a finite number. by the logarithmic method ( L) if converges for 0 < x < 1 and tends to. as x. 1 -. Our main object in this paper is to introduce new Tauberian conditions of the " one-sided bounded" or " slowly decreasing" type to retrieve ordinary convergence of a real sequence from its (L) summability.