  In this paper, we prove the uniqueness of solutions to the logarithmic
Minkowski problem in $\mathbb{R}^3$ without symmetry condition, provided the
density of the measure is close to $1$ in $C^{\alpha}$ norm. This result also
implies the uniqueness of self-similar solutions to the anisotropic Gauss
curvature flow in $\mathbb{R}^3$ when the speed function is $C^{\alpha}$ close
to a positive constant.
