Summary: We study continued logarithms, as introduced by Gosper and studied by Borwein et al. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchin's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchin constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize beyond continued logarithms.

11J70, 11K50

continued fraction generalization, continued logarithm, Khinchin constant