In this article, we investigate the construction of a meshless local discrete collection method suitable for solving a class of boundary integral equations of the second kind with logarithmic singular kernels. These types of boundary integral equations can be deduced from boundary value problems of Laplace's equations with linear Robin boundary conditions. The numerical solution presented in the current paper is obtained by moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The logarithm-like singular integrals appeared in the method are computed via a particular nonuniform Gauss-Legendre quadrature rule. Since the offered scheme is based on the use of scattered points spread on the solution domain and does not require any background meshes, it can be identified as a meshless local discrete collocation (MLDC) method. We also obtain the error bound and the convergence rate of the presented method. The new technique is simple, efficient and flexible for most classes of boundary integral equations. The convergence accuracy of the new technique is examined over four integral equations on various domains and obtained results confirm the theoretical error estimates.