  In this paper we investigate the stability of a recently introduced
Bose-Einstein condensate (BEC) which involves logarithmic interaction between
atoms. The Gaussian variational approach is employed to derive equations of
motion for condensate widths in the presence of a harmonic trap. Then we derive
the analytical solutions for these equations and find them to be in good
agrement with numerical data. By analyzing deeply the frequencies of collective
oscillations, and the mean-square radius, we find that the system is always
stable for both negative and positive week logarithmic coupling. However, for
strong interaction the situation is quite different: our condensate collapses
for positive coupling and oscillates with fixed frequency for negative one.
These special results remain the most characteristic features of the
logarithmic BEC compared to that involving two-body and three-body
interactions.
