  We study the logarithmic Schr\"odinger equation with finite range potential
on $\mathbb{R}^{\mathbb{Z}^d}$. Through a ground-state representation, we
associate and construct a global Gibbs measure and show that it satisfies a
logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary
dimension and prove the existence of weak solutions to the infinite-dimensional
Cauchy problem.
