  Using a spherically symmetric ansatz, we show that the Chern-Simons
O(3)-sigma model with a logarithmic potential admits topological solutions.
This result is quite interesting since the Gausson-type logarithmic potential
only predicted topological solutions in $(1+1)$D models. To accomplish our
goal, the Bogomol'nyi-Prasad-Sommerfield (BPS) method is used, to saturate the
energy and obtain the BPS equations. Next, we show by the numerical method is
the graphical results of the topological fields, as well as, the magnetic field
behavior that generates a flux given by $\Phi_{flux}=-\mathcal{Q}/\kappa$ and
the energy density of the structures of vortices. On the other hand, we
evaluate the measure of the differential configurational complexity (DCC) of
the topological structures, by considering the energy density of the vortex.
This analysis is important because it will provide us with information about
the possible phase transitions associated with the localized structures and it
shows that our model only supports the one-phase transition.
