  We analyze some issues related to the stability and free expansion of a
one-dimensional logarithmic Bose-Einstein condensate, particularly its eventual
relation to the formation of quantum droplet-type configurations. We prove that
the corresponding properties, such as the energy of the associated N-body
ground state, differ substantially with respect to its three-dimensional
counterpart. Consequently, the free velocity expansion also shows differences
with respect to the three-dimensional system when logarithmic interactions are
taken into account. The one-dimensional logarithmic condensate tends to form
quantum droplet-type configurations when the external trapping potential is
turned off, i.e., the self-sustainability or self-confinement appears as in
three-dimensions. However, we obtain that for some specific values of the
self-interaction parameters and the number of particles under consideration,
the cloud oscillates during the free expansion around to a specific equilibrium
size. These results show that we can get scenarios in which the one-dimensional
cloud reaches stable configurations, i.e., oscillating quantum droplets.
