不定对数、对数单位和熵的性质
The Indefinite Logarithm, Logarithmic Units, and the Nature of Entropy
摘要：
我们将实数x>0的不定对数[log x]定义为表示具有不定底的x的对数的抽象概念的数学对象（即，不具体是E、10、2或任何固定数）。由此产生的不确定对数量自然地扮演着一种数学角色，这种角色与维度物理量（例如长度）的角色非常相似，因为尽管这些量没有作为普通数的明确解释，但是这些实体中的两个的比率自然地被定义为特定的普通数，就像两个长度的比率一样。因此，不确定的对数对象可以作为对数空间的基础，对数空间是对数单位的自然系统，适用于测量在对数尺度上定义的任何量。我们说明了对数单位如何提供一种方便的语言，用于解释目前用于各种传统上被认为是不同的量，如，特别是物理熵和信息论熵的不同系统的单位的完整的概念统一。
英文摘要：
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.
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