We present new Tauberian conditions in terms of the general logarithmic control modulo of the oscillatory behavior of a real sequence (sn) to obtain lim n→∞ sn = ξ from st − limn→∞ sn = ξ, where ξ is a finite number. We also introduce the statistical (`,m) summability method and extend some Tauberian theorems to this method. The main results improve some well-known Tauberian theorems obtained for the statistical convergence.