In this paper, we prove several Tauberian remainder theorems on \(\lambda \) -bounded sequences with respect to iterations of the logarithmic summability method. We establish the relationship between the general logarithmic control modulo and the logarithmic mean of a sequence. Additionally, we present an equality between the general logarithmic control modulo and the generator sequence. In the main results of this study, the \(\lambda \) -boundedness of a sequence is obtained with the help of some conditions on the general logarithmic control modulo and the generator sequence. Furthermore, the results in this work generalize some Tauberian remainder theorems previously established for the logarithmic method of summability.
