Indefinite logarithm
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The indefinite logarithm of a positive number n (variously denoted [logn], Log(n) or even sometimes just logn) is just the logarithm with respect to any base, when we don't worry about the base. This is as opposed to the ordinary, or definite logarithm, where there is always (implicitly or explicitly) a particular base to which the logarithm is being taken.
As an example, if we want to know which power of 2 is 256, we just compute log(256)/log(2); this will give the answer 8 regardless of which logarithm function we used (as long as we use the same one for log(256) and log(2)).
In other words, an indefinite logarithm is a function that is known to have the properties of any logarithm function (i.e., it is defined for all x > 0, and log(1)=0, and log(ab)=log(a)+log(b)), but where we just don't know which base b satisfies log(b)=1, and we don't need to know.
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[edit] Definition
The indefinite logarithm operator Log can be defined as the unary operator such that, for any given x > 0, Log(x) returns the entire logarithmic function object 
, which itself maps any given base b > 0 to the logarithm of x base b. Using lambda calculus notation, we can express this definition of the Log operator a bit more formally as Log = λx.(λb.logb(x)). With this definition, one can easily define addition of indefinite logarithms and their multiplication by scalars, thereby forming a complete vector space of indefinite logarithm quantities.
One way to understand the meaning of the indefinite logarithm is to think of it as a dimensioned (i.e., not dimensionless) quantity. Any such quantity is expressible (in infinitely many ways) as a pair of a (dimensionless) pure number and an arbitrary unit quantity, analogously to how we express dimensioned physical quantities, such as length, time, or energy (See dimensional analysis). In the case of the quantities that result from the indefinite logarithm function, we may call their associated units logarithmic units. Logarithmic units are themselves indefinite-logarithm quantities, and can be represented with the same notation, e.g., ![[\log\,n]](../../../math/1/c/f/1cf29b79df571248ceaff2746a798fe1.png)
for the logarithmic unit which is equal to the indefinite logarithm of n.
[edit] In physics
In physics, two units of the same physical dimensions generally have a well-defined numerical ratio between them, such as, for example, (1 in)/(1 cm) = 2.54. Similarly, two indefinite logarithmic units [loga] and [logb] have a definite numerical ratio between them, given by [loga] / [logb] = logba. This follows because logca / logcb has always the same value, namely logba, regardless of what particular numerical base c > 0 we might choose as the base of our logarithms.
Thus, replacing the indefinite logarithm by a definite logarithm can be compared to representing a length or other physical quantity using a specific unit of measurement. In some contexts, the "unit" for logarithms base 10 are called "bel", abbreviated B and most commonly encountered as decibel, dB. Similarly, logarithms base 2 are sometimes called "bit", base 256 "byte", and base e "napier".
[edit] In general
In general, the same identities hold for indefinite logarithms as hold for ordinary logarithms (with a given consistent choice of base).
We can also define an indefinite exponential, denoted Exp(L), which is well-defined (with a pure-number value n) for indefinite-logarithm quantities L = Log(n).
The concepts of indefinite logarithms (and indefinite exponentials) are useful when discussing physical or mathematical quantities that are most naturally defined in terms of logarithms, such as (in particular) information and entropy. Such quantities can be considered to be most naturally expressed in terms of indefinite logarithms; that is, their natural units are (or involve) logarithmic units.
