This correspondence establishes an interesting connection between the Kullback-Leibler divergence and the Perron root of nonnegative irreducible matrices. In the second part of the correspondence, we apply these results to the power control problem in wireless communications networks to show a fundamental tradeoff between fairness and efficiency. A power vector is said to be efficient if it maximizes the overall network efficiency expressed in terms of an aggregate network utility function parameterized by some weight vector. For two widely used examples of utility functions, the correspondence identifies the unique weight vector for which a power vector is both efficient and max-min fair in the sense that each communication link has the same quality-of-service. These results also give rise to new saddle point characterizations of the Perron root