  In this paper, we introduce a conservative Crank-Nicolson-type finite
difference schemes for the regularized logarithmic Schr\"{o}dinger equation
(RLSE) with Dirac delta potential in 1D. The regularized logarithmic
Schr\"{o}dinger equation with a small regularized parameter $0<\eps \ll 1$ is
adopted to approximate the logarithmic Schr\"{o}dinger equation (LSE) with
linear convergence rate $O(\eps)$. The numerical method can be used to avoid
numerical blow-up and/or to suppress round-off error due to the logarithmic
nonlinearity in LSE. Then, by using domain-decomposition technique, we can
transform the original problem into an interface problem. Different treatments
on the interface conditions lead to different discrete schemes and it turns out
that a simple discrete approximation of the Dirac potential coincides with one
of the conservative finite difference schemes. The optimal $H^1$ error
estimates and the conservative properties of the finite difference schemes are
investigated. The Crank-Nicolson finite difference methods enjoy the
second-order convergence rate in time and space. Numerical examples are
provided to support our analysis and show the accuracy and efficiency of the
numerical method.
