Low-rank matrix approximation finds wide application in the analysis of big data, in recommendation systems on the Internet, for approximate solution of some equations in mechanics, and in other fields. In the paper, a method for approximating positive matrices by matrices of unit rank is proposed on the basis of minimization of log-Chebyshev distance. The approximation problem is reduced to an optimization problem that has a compact representation in terms of an idempotent semifield that uses the operation of taking maximum in the role of addition, and is often called the max-algebra. The necessary definitions and preliminary results of tropical mathematics are given, on the basis of which the solution of the original problem is derived. Using methods and results of tropical optimization, all the positive matrices which provide the minimum of approximation error are obtained in explicit form. A numerical example that illustrates the application of the proposed rank-one approximation method is considered.