A irracionalidade e transcendência de certos logaritmos

In this article we show some demonstrations of irrationality of certain logarithms. The importance of this fact is related to the characterization of the irrational numbers by the infinite non-periodic decimal representation. In both the old logarithmic tables and the electronic calculators, the logarithmic values are represented in the form of a finite decimal expression. This may generate the false impression that they are necessarily rational numbers. Unlike many texts and articles on the subject, we are not restricted to the decimal logarithm. In fact, we show a sufficient condition for the log b number to be irrational, where a > 0 and b > 1 are integers and also a criterion that establishes the irrationality of log b when b > 1 is square free. We finally show, using GelfondSchneider’s Theorem, some proof of the transcendence of certain logarithms, and we deal with logarithms of non-integer real numbers.