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.