The logarithmic Schrödinger equation is a partial differential equation. In mathematics and mathematical physics one often uses its dimensionless form: i ∂ ∂ ψ ψ ∂ ∂ t + ∇ ∇ 2 ψ ψ + ψ ψ ln ⁡ ⁡ | ψ ψ | 2 = 0. for the complex-valued function ψ = ψ (x, t) of the particles position vector x = (x, y, z) at time t, and ∇ ∇ 2 ψ ψ = ∂ ∂ 2 ψ ψ ∂ ∂ x 2 + ∂ ∂ 2 ψ ψ ∂ ∂ y 2 + ∂ ∂ 2 ψ ψ ∂ ∂ z 2 is the Laplacian of ψ in Cartesian coordinates. The logarithmic term ψ ψ ln ⁡ ⁡ | ψ ψ | 2 has been shown indispensable in determining the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures. This logarithmic term is also needed for cold sodium atoms. In spite of the logarithmic term, it has been shown in the case of central potentials, that even for non-zero angular momentum, the LogSE retains certain symmetries similar to those found in its linear counterpart, making it potentially applicable to atomic and nuclear systems.