As a consequence of the methods developed in the earlier papers of this series [1, 2, 3], an effective algorithm has recently been established for solving many Diophantine equations in two unknowns (see [4,5,6,7]). The algorithm leads to an explicit bound for the size of all the solutions, and, in principle therefore, it enables any specific equation of the type considered to be fully resolved by a finite amount of computation. On examining the various estimates occurring in the course of the exposition, however, it at once became apparent that the computation would involve a very large number of operations and would scarcely be practicable even with a modern machine. It was clear, on the other hand, that a modified version of the fundamental inequality involving the logarithms of algebraic numbers would much facilitate the computational work, and it was in the light of this observation that the researches discussed herein were begun. The object has been to obtain a theorem of an essentially practical nature which may be found useful in application to a wide variety of different problems. The result which we shall establish is neither the most precise nor the most general that can be obtained in this direction, but it would seem to be the most serviceable of its kind, and it would apparently make feasible many calculations which would otherwise have seemed quite out of the question.