The tanh rule and the double-exponential (DE) formula are known empirically and theoretically as quite efficient quadrature formulas, especially for integrals with endpoint singularity, including algebraic singularity and logarithmic singularity. Furthermore, in the case of integrals with algebraic singularity, explicit error bounds have been given for those formulas, which enables us to guarantee their approximation accuracy. In the case of integrals with logarithmic singularity, however, such explicit error bounds have not ever given thus far, although those formulas should work accurately in this case as well. This paper presents the desired theoretical explicit error bounds, with numerical experiments.