The logarithm of irrational numbers and Beatty sequences

Throughout, we use [x] to represent the greatest integer not exceeding x, and define {x} := x - [x]. Unless otherwise specified, the letters a and b represent irrational numbers. Also, A and B represent the Beatty sequences A := ([an])(n=1)(infinity) and B := ([bn])(n=1)(infinity) . We will often refer to integers when we really mean positive integers, but this will not generate any conceptual ambiguity.