We present a fast Galerkin spectral method to solve logarithmic singular equations on segments.The proposed method uses weighted first-kind Chebyshev polynomials.Convergence rates of several orders are obtained for fractional Sobolev spaces H~-1/2 (or H00-1/2).Main tools are the approximation properties of the discretization basis,the construction of a suitable Hilbert scale for weighted L2-spaces and local regularity estimates.Numerical experiments are provided to validate our claims.