In the paper, we introduce two new classes of orthogonal functions, new log orthogonal functions (NLOFs) and generalized new log orthogonal functions (GNLOFs), which are based on generalized Laguerre polynomials. We construct basis approximations analysis for the new orthogonal functions and apply them to solve several typical differential equations. Our errors analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions which have weak singularities at one endpoint, and can obtain exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.