The paper contains the proof of theorem estimating the condition number of matrix after discretization of the integral equation of the first kind with logarithmic singularity on the closed curve by the collocation method with piecewise-linear approximation of unknown function. It is based on a new approach using convex properties of images of the basic functions, so it gives the opportunity to study a quasi-uniform sequence of grids. If the initial integral equation is uniquely solvable, the quasiuniform parameter does not exceed 4.25 and the step is sufficiently small, then the norm of finite-dimensional inverse operator is not greater than a constant divided by the step.