We develop a class of conservative integrators for the regularized logarithmic Schrödinger equation (RLogSE) using the quadratization technique and symplectic Runge-Kutta schemes. To preserve the highly nonlinear energy functional, the regularized equation is first transformed into an equivalent system that admits two quadratic invariants by adopting the invariant energy quadratization approach. The reformulation is then discretized using the Fourier pseudo-spectral method in the space direction, and integrated in the time direction by a class of diagonally implicit Runge-Kutta schemes that conserve both quadratic invariants to round-off errors. For comparison purposes, a class of multi-symplectic integrators are developed for RLogSE to conserve the multi-symplectic conservation law and global mass conservation law in the discrete level. Numerical experiments illustrate the convergence, efficiency, and conservative properties of the proposed methods.