The asymptotic behavior of several sequence transformations is investigated as n -► oo when applied to a certain family of logarithmically convergent sequences. The transformations considered are the iterations of the transformations e^\An) of Shanks and Wn of Lubkin, the ^-algorithm of Brezinski, the Levin u-and u-transforms, and generalizations of the palgorithm and the Neville table. Computational results are given for both real and complex sequences.