Logarithmic units

Logarithmic units are abstract mathematical units that can be used to express any quantities (physical or mathematical) that are defined on a logarithmic scale, that is, as being proportional to the value of a logarithm function. In this article, a given logarithmic unit will be denoted using the notation , where n is a positive real number, and here denotes the indefinite logarithm function Log().

Related Topics:
Logarithmic scale - Logarithm - Indefinite logarithm

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Examples of logarithmic units include common units of information, such as the bit and the byte 8 = ; units of entropy such as the nat ; units of relative signal strength magnitude such as the decibel 0.1; and other logarithmic-scale units such as the Richter scale point or (more generally) the corresponding order-of-magnitude unit sometimes referred to as a factor of ten or decade (here meaning , not 10 years).

Related Topics:
Information - Bit - Byte - Entropy - Decibel - Richter scale

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The motivation behind the concept of logarithmic units is that defining a quantity on a logarithmic scale 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 (and equally arbitrary) logarithm base that was selected. Due to the identity log_b,a = (log_c,a)/(log_c,b), the logarithms of any given number a to two different bases (here b and c) differ only by the constant factor logc b. This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity Log(a) from one arbitrary unit of measurement (the unit) to another (the unit), since mathrm{Log}(a) = (log_b,a) = (log_c,a).

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For example, Boltzmann's standard definition of entropy S = k ln W (where W is the number of ways of arranging a system and k is Boltzmann's constant) can also written more simply as just S = Log(W), where "Log" here denotes the indefinite logarithm, and we let k = ; that is, we identify the physical entropy unit k with the mathematical unit . This identity works because ln,W = log_e,W = mathrm{Log}(W)/. Thus, we can interpret Boltzmann's constant as being simply the expression (in terms of more standard physical units) of the abstract logarithmic unit that is needed to convert the dimensionless pure-number quantity ln W (which uses an arbitrary choice of base, namely e) to the more fundamental pure logarithmic quantity Log(W), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.

Related Topics:
Boltzmann - Boltzmann's constant - Indefinite logarithm

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