A novel finite basis set method is used to calculate the Bethe logarithm for the ground 2 (2)S(1/2) and excited 3 (2)S(1/2) states of lithium. The basis sets are constructed to span a huge range of distance scales within a single calculation, leading to well-converged values for the Bethe logarithm. The results are used to calculate an accurate value for the complete quantum electrodynamic energy shift up to order alpha(3) Ry. The calculated 3 (2)S(1/2)-2 (2)S(1/2) transition frequency for 7Li is 27 206.092 6(9) cm(-1), and the ionization potential for the 2 (2)S(1/2) state is 43 487.158 3(6) cm(-1). The 7Li-6Li isotope shift is also considered, and all the results compared with experiment.