Charged black points in general relativity coupled to the logarithmic U(1) gauge theory

The exact solution for a static spherically symmetric field outside a charged point particle is found in a non-linear U(1) gauge theory with a logarithmic Lagrangian. The electromagnetic self-mass is finite, and for a particular relation between mass, charge, and the value of the non-linearity coupling constant, \lambda, the electromagnetic contribution to the Schwarzschild mass is equal to the total mass. If we also require that the singularity at the origin be hidden behind a horizon, the mass is fixed to be slightly less than the charge. This object is a {\em black point.}