  We study continued logarithms as introduced by Bill Gosper and studied by J.
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 Khinchine's constant
from simple continued fractions to continued logarithms, and show that these
logarithmic Khinchine 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 past continued
logarithms.
