Definitions

# Indefinite logarithm

The indefinite logarithm of a positive number $n$ (variously denoted $\left[log n\right]$, $mathrm\left\{Log\right\}\left(n\right)$ or even sometimes just $log n$) is the logarithm without regard to any particular base: it is a function (of the base), not a number. 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.

In other words, an indefinite logarithm of a number is a function that is known to have the properties of any logarithm function (i.e., it is defined for all $x>0$, $log 1=0$, and $log ab=log a + log b$), where the base is unknown and the knowledge of the base of the logarithm is unnecessary: it defers the choice of base.

## Definition

The indefinite logarithm operator $mathrm\left\{Log\right\}$ can be defined as the unary operator such that, for any given $x > 0$, $mathrm\left\{Log\right\}\left(x\right)$ returns the entire logarithmic function object $b mapsto log_b\left(x\right)$, 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 $mathrm\left\{Log\right\} = lambda x.\left(lambda b.log_b\left(x\right)\right)$. 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, analogous to the expression of 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, their associated units are called logarithmic units. Logarithmic units are themselves indefinite-logarithm quantities, and can be represented with the same notation, e.g., $\left[log n\right]$ for the logarithmic unit which is equal to the indefinite logarithm of $n$.

### Mathematical details

The logarithm $log_b x$ is a function in two variables: the base b and the argument x. If one fixes the base, one obtains the definite logarithm, which is an function of x. If one fixes the number x, one obtains the indefinite logarithm, which is a function of the base b. That is,
$operatorname\left\{Log\right\}\left(x\right) = frac\left\{log x\right\}\left\{log b\right\}$
Note that the definite logarithm is an increasing function of the argument x, while the indefinite logarithm is a decreasing function of the base: as the size of units increase, the value of the logarithm for a fixed argument decreases.

Converting a function in two variables into a function of one variable by fixing one of the arguments is known as currying.

Similarly, given a dimensional quantity such as length, one converts it to a dimensionless number by dividing by a unit: Length(x) = length(x)/length(b): the larger your unit, the smaller your value in those units.

## 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 $\left[log a\right]$ and $\left[log b\right]$ have a definite numerical ratio between them, given by $\left[log a\right] / \left[log b\right] = log_b a$. This follows because $log_c a / log_c b$ has always the same value, namely $log_b a$, 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 "neper".

## 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 $mathrm\left\{Exp\right\}\left(L\right)$, which is well-defined (with a pure-number value $n$) for indefinite-logarithm quantities $L = mathrm\left\{Log\right\}\left(n\right)$.

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, they take a value on a logarithmic scale, though there may not be a natural choice of logarithmic units.