[en] Conformal invariance in one-dimension implies the correlation functions must be constant. In this paper, it is demonstrated by an exact solution that all the correlations between like species in a two-component lattice gas with the logarithmic potential have the conformal invariance property at the metal- insulator transition. A further exactly solvable isotherm of the model system is studied and the corresponding density of zeros for the grand partition function is obtained explicitly. A phase transition along this isotherm can be induced by appropriate choice of the parameters