Low-rank matrix approximation is widely used in the analysis of big data, in recommendation systems in the Internet, for approximation solution of some equations in mechanics, and other fields. In many applications it makes sense to use matrices of unit rank for approximating since they have the simplest structure. This article provides a method for approximating positive matrices by matrices of unit rank based on the minimization of log-Chebyshev distance. The approximation problem is reduced to the optimization problem, which has a compact representation in terms of an idempotent semifield that taking maximum in the role of addition. Such semifield is often called the max-algebra. The necessary definitions and results of tropical mathematics are given and the solution of the optimization problem is derived from them. Then the solution is represented in terms of the original approximation problem. As a result, all the positive matrices which provide the minimum of approximation error are obtained in explicit form.