We construct two new classes of log orthogonal functions in semi-infinite intervals, log orthogonal functions (LOFs-II) and generalized log orthogonal functions (GLOFs-II), by applying a suitable log mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and show that they can provide uniformly good exponential convergence rates for problems in semi-infinite intervals with slow decay at infinity. We apply them to solve several linear and nonlinear differential equations whose solutions decay algebraically or exponentially with very slow rates, and present ample numerical results to show the effectiveness of the approximations by LOFs-II and GLOFs-II.