Logarithmic units are generic mathematical units in which we can express any quantities (physical or mathematical) that are defined as being proportional to values of a logarithm function. Here, a given logarithmic unit will be denoted using the notation , where n is a positive real number, and "log" here denotes the indefinite logarithm function. In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. ... The indefinite logarithm (sometimes denoted or ) of a positive number is a special type of mathematical object representing the abstract concept of a logarithm of This is as opposed to the ordinary, or definite logarithm, in which there is always (implicitly or explicitly) a particular base to which the logarithm...

Examples of logarithmic units include common units of information, such as the bit and the byte ; units of entropy such as the nat ; units of relative signal strength magnitude such as the decibel ; and other logarithmic-scale units such as the Richter scale point or the generic order-of-magnitude unit sometimes referred to as the factor of ten or decade or (here meaning , not 10 years). Information is a term with many meanings depending on context, but is as a rule closely related to such concepts as meaning, knowledge, instruction, communication, representation, and mental stimulus. ... The thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ... The Richter magnitude test scale (or more correctly local magnitude ML scale) assigns a single number to quantify the size of an earthquake. ...

The motivation behind the concept of logarithmic units is that defining a quantity in terms of a logarithm to a specific base amounts to making a (totally arbitrary) choice of a unit of measurement for that quantity, one that corresponds to the specific logarithm base that was (equally arbitrarily) chosen. Due to the identity , the logarithms of any number a to two different bases (here b and c) differ only by the constant factor . This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity from one arbitrary unit of measurement (the unit) to another (the unit).

For example, Boltzmann's standard definition of entropy , (where W is the number of ways of arranging a system and k is Boltzmann's constant) can also written more simply as , where "log" here denotes the indefinite logarithm, and we let . This works because of the property of the indefinite logarithm that W]/[log e]</math>. Thus, we can interpret Boltzmann's constant as being simply the expression (in standard physical units) of the logarithmic unit that is needed to convert the dimensionless pure-number quantity (which uses an arbitrary choice of base, namely e) to the more fundamental pure logarithmic quantity , which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy. The indefinite logarithm (sometimes denoted or ) of a positive number is a special type of mathematical object representing the abstract concept of a logarithm of This is as opposed to the ordinary, or definite logarithm, in which there is always (implicitly or explicitly) a particular base to which the logarithm...