An Essay on Irrationality Measures of Logarithms

produces ‘good’ rational approximations to log a. There are several ways of perfoming integration in (1) in order to show that the integrals lies in Q log a + Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and log 3. Although methods presented below work in general situations (e.g., for certain Q-linear forms in logarithms) as well, the three numbers seem to be very nice and important models for our exposition. Bounds for irrationality measures are presented by means of upper estimates for irrationality exponents. Recall that the irrationality exponent of a real irrational number γ is defined by the relation