  We define the indefinite logarithm [log x] of a real number x>0 to be a
mathematical object representing the abstract concept of the logarithm of x
with an indeterminate base (i.e., not specifically e, 10, 2, or any fixed
number). The resulting indefinite logarithmic quantities naturally play a
mathematical role that is closely analogous to that of dimensional physical
quantities (such as length) in that, although these quantities have no definite
interpretation as ordinary numbers, nevertheless the ratio of two of these
entities is naturally well-defined as a specific, ordinary number, just like
the ratio of two lengths. As a result, indefinite logarithm objects can serve
as the basis for logarithmic spaces, which are natural systems of logarithmic
units suitable for measuring any quantity defined on a logarithmic scale. We
illustrate how logarithmic units provide a convenient language for explaining
the complete conceptual unification of the disparate systems of units that are
presently used for a variety of quantities that are conventionally considered
distinct, such as, in particular, physical entropy and information-theoretic
entropy.
