  We investigate the presence of vortex structures in a Maxwell model with a
logarithmic generalization. This generalization becomes important because it
generates stationary field solutions in models that describe the dynamics of a
scalar field. In this work, we will choose to investigate the dynamics of the
complex scalar field with the gauge field governed by Maxwell term. For this,
we will investigate the Bogomol'nyi equations to describe the static field
configurations. Then, we show numerically that the complex scalar field
solutions that generate minimum energy configurations have internal structures.
Finally, assuming a planar vision, the magnetic field and the density energy
show the interesting feature of the ringlike vortex.
