  We present two new classes of orthogonal functions, log orthogonal functions
(LOFs) and generalized log orthogonal functions (GLOFs), which are constructed
by applying a $\log$ mapping to Laguerre polynomials. We develop basic
approximation theory for these new orthogonal functions and apply them to solve
several typical fractional differential equations whose solutions exhibit weak
singularities. Our error 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 lead to exponential
convergence rate, as opposed to low algebraic rates if usual orthogonal
polynomials are used.
