In this article, we mainly study the global existence of multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb type potential

$$-\mathrm{\Delta }u+V(ϵx)u=\lambda (I_{\alpha }\ast |u{|}^p)|u{|}^{p-1}+u\log u^2\text{in}{\mathbb{R}}^3,$$

where $u\in H^1({\mathbb{R}}^3)$, $ϵ>0$, $V$ is a continuous function with a global minimum, and Coulomb type energies with $0<\alpha <3$ and $p\ge 1$. We explore the existence of local positive solutions without the functional having to be a combination of a $C^1$ functional and a convex semicontinuous functional, as is required in the global case.