I shall describe any homoorphism from a multiplication to an addition
as logarithmic
. I shall begin by discussing a function
whose outputs are (when mappings) logarithmic from the multiplication on
{positives} to the addition on {reals}, but discussed in a generality that
allows their application wherever power's definition
can sensibly be applied or extended. The general properties
of such mappings from {positives} to {reals} then prepare the ground for the
introduction of the natural logarithm, in terms of which
all other logarithmic functions may be expressed.
Let log = reverse&on;transpose(power). This is
so log(x, v) = n precisely if power(n, x) = v, and thus power(log(x, v), x) = v for all positive x ≠ 1 and v. (This is xlogx(v) = v in orthodox notation.) As power(0, x) = 1 and power(1, x) = x, log(x) relates 0 to 1 and 1 to x, i.e. log(x, 1) = 0 and log(x, x) = 1, for all x.
As power(x, 1) = 1 for all x, log(1) relates every input power can accept to 1; it has only one right value and it is not a mapping. As power(0, 0) = 1, log(0) does relate 0 to 1 (and can be thought of as mapping 1 to 0); however power(x, 0) = 0 for all positive x, so log(0) relates all positives to 0 (and doesn't relate anything else to anything else); as a result, it isn't a mapping. So, while log(0) and log(1) are well-defined as relations, they're not mappings.
As transpose(power, x) is strictly monotonic (increasing for x > 1, decreasing for 0 < x < 1) for all positive x other than 1, log(x) is a mapping when either 0 < x < 1 or 1 < x. As power(−n, x) = 1/power(n, x) = power(n, 1/x), log(1/x) = −log(x), so everything of interest about log(x) for 0 < x < 1 can be inferred from log(z) for some z > 1. Thus I shall attend primarily to log(x) for x > 1.
Consider log(x, u.v), with power(log(x, u.v), x) = u.v = power(log(x, u), x).power(log(x, v), x); and each (: power(t, x) ←t :) is exponential so this last is power(log(x, u) +log(x, v), x); as long as log(x) is a mapping (i.e. (: power(t, x) ←t :) is monic), this makes log(x, u.v) = log(x, u) +log(x, v), so that each log(x) that's a mapping is a homomorphism from a multiplication to an addition; every mapping that's an output of log is logarithmic (and this, of course, is the reason for its name).
Now, power(log(x, v), x) = v = power(log(y, v), y), so power(log(x, v)/log(y, v), x) = y = power(log(x, y), x) whence log(x, v)/log(y, v) = log(x, y) at least in so far as (: power(u, x) ←u :) is monic, i.e. log(x) is a mapping. This gives us log(x, v) = log(x, y).log(y, v) for all x, y, v; whence log(x) = log(x, y).log(y) as functions, for all x, y. In particular, 1 = log(x, x) = log(x, y).log(y, x), i.e. log(y, x) = 1/log(x, y), whenever log(x) and log(y) are mappings, in particular whenever x and y are positives other than 1. When x has a multiplicative inverse, log(x) relates −1 to 1/x; when log(x) is a mapping, this gives log(1/x, y) = log(1/x, x).log(x, y) = −log(x, y) and, when log(y) is also a mapping, log(y, 1/x) = −log(y, x).
Consider any logarithmic f; i.e. a homomorphism from a multiplication to an addition, f(x.y) = f(x) +f(y) and f(1) = 0. If we scale f by a constant, k.f(x.y) = k.(f(x) +f(y)) = k.f(x) +k.f(y), at least when our multiplication distributes over our addition (e.g. when f's outputs and k are values of some ringlet), making k.f also logarithmic for any k. When the outputs of f lie in some module over a ringlet, e.g. in a vector space, selecting a single component of it respects addition, so the composite of such a selection after f will be logarithmic when f is; for example, the real (or imaginary) part of a complex-valued logarithmic function.
As long as logarithmic ({reals}: f :) isn't (everywhere) zero, there's some x for which f(x) is non-zero, whence there is some integer n for which 1 < n.f(x) = f(power(n, x)) and so there is some input, u = power(n, x), for which f(u) > 1; and thus, for every natural m, there is some natural for which m < m.f(u) = f(power(m, u)) and so there is no upper bound, in {naturals}, on the outputs of f; likewise, by taking the multiplicative inverses of inputs with large positive output, we get inputs that f maps to arbitrarily large negative values; there is no bound on the range of f's outputs, unless f is zero.
Furthermore, any logarithmic (: f |{positives}) and any positive x with f(x) non-zero give us, for every positive natural n, power(1/n, x) which f maps to f(x)/n, whence we have inputs that f maps to non-zero values arbitrarily close to zero; and we can subdivide the range between two outputs, f(x) and f(r.x), arbitrarily finely with f(x.power(i/n, r)) for various i and n to establish that any logarithmic (: |{positives}) is continuous.
As any non-zero logarithmic ({reals}: :) has unbounded range, we can infer that any non-zero ({reals}: f |{positives}) is in fact ({reals}| f |{positives}), so every real k is f(v) for some v.
Now, for a logarithmic (: f |{positives}), suppose we're given x, y for which f(x) = f(y); thus f(x/y) = f(x) −f(y) = 0; thus f(power(n, x/y)) = n.f(x/y) = 0 for every integer n. If x and y are distinct, this gives us infinitely many inputs (forming a geometric sequence) that f maps to zero. If a positive r has power(n, r) = power(m, x/y) for any non-zero integral n, m, then n.f(r) = m.f(x/y) = 0, so f(r)'s order, in the output addition, is a divisor of (non-zero) n. That can be realised by f(r) = 0 with order 1, which trivially divides all n; otherwise, it can only happen if the addition has some non-zero values with non-zero order. Thus a logarithmic(: f |{positives}), whose output addition (e.g. real addition) has order zero for all non-zero values, takes value zero at every rational power of any input at which its output is zero. Thus, if a logarithmic function ({reals}: |{positives}) isn't monic, it has zero output at a dense sub-set of {positives} and continuity obliges it to be zero everywhere; so any non-zero logarithmic ({reals}: f |{positives}) is monic and, thanks to the orderings on {reals} and {positives}, necessarily strictly monotonic.
Now consider two homomorphisms from positive multiplication to real addition, logarithmic f, g each ({reals}: |{positives}). If they're both zero, then they're equal and, in particular, either of them is the other scaled by one of its outputs. Otherwise, one of them is non-zero and thus ({reals}| :{positives}), continuous and monic; suppose this is f. Since f(1) = 0 and f is monic, any x other than 1 has non-zero f(x) and we can divide g(x)/f(x). With y = power(r, x) for any rational r, we obtain g(y) = g(power(r, x)) = r.g(x) = r.f(x).(g(x)/f(x)) = f(power(r, x)).(g(x)/f(x)) = f(y).g(x)/f(x). Thus g(y) = f(y).g(x)/f(x) holds true for all y in a dense sub-set of {positives} and continuity of the expressions it equates requires it to hold true for every positive y. Consequently, given any non-zero logarithmic ({reals}: f |{positives}), any logarithmic ({reals}: g |{positives}) can be expressed as k.f for some constant real k; and this k is necessarily an output of f.
In particular, log(x) is logarithmic ({reals}: |{positives}) for every positive x ≠ 1; hence any logarithmic g maps each y to log(x, y).g(u)/log(x, u) for any positive u ≠ 1; and log(x, y)/log(x, u) = log(u, y), so g(y) = g(u).log(u, y) or g = g(u).log(u) for any positive u ≠ 1. If g(u) is 0, in such a case, then g is zero; otherwise, 1/g(u) is an output of log(u), i.e. 1/g(u) = log(u, v) for some v; whence g(u) = log(v, u) and g(y) = log(v, u).log(u, y) = log(v, y) so g = log(v). Thus any logarithmic mapping is either zero (and boring) or an output of log. Furthermore, every non-zero logarithmic g, by being log(v) for some v, gives us log(x, y) = log(v, y)/log(v, x) = g(y)/g(x) so generates log = (: g/g(x) ←x :). We only ever need one logarithmic function to describe all logarithmic functions.
I'll now derive a non-zero logarithmic function known as
the natural logarithm
and show how it produces some
natural results, which shall give us a new way to understand (and generalise)
the extension of power. The natural logarithm's orthodox
name, ln
, is an abbreviation
of logarithm natural
, probably originally in some other
language where adjectives follow the nouns they apply to.
The positive reals form a multiplicative group with a cancellable addition, whose completion will give us the reals. In the two-dimensional quadrant {lists ({positives}: |2)}, the set {[x, y]: x.y = 1} depicts the multiplicative inversion function, power(−1). This curve is invariant under a rescaling of the quadrant that scales each co-ordinate independently, provided the product of the factors is 1; thus, for any positive c, {[c.x, y/c]: x.y = 1} is the same curve. Let us now consider the area of any region bounded by: this curve, an edge of the quadrant and any pair of lines parallel to the other edge of the quadrant, at distances a and b from that edge, with a < b. Let the area thus enclosed by denoted A(a, b) for now. Since juxtaposing two sutch strips between the curve and the axis produces a wider strip, A(a, b) +A(b, c) = A(a, c) for all positive a, b, c with a < b < c. Note, by observing a rectangle inside the area and a rectangle that contains it, that (b−a)/b < A(a, b) < (b−a)/a.
Indeed, we can refine this bound: first consider any point [x, y] on the chord from [a, 1/a] to [b, 1/b]; it has
which is necessarily >0 for a < x < b, hence x.y > 1 and y > 1/x on the interior of the chord, which thus lies above the curve, hence the area under the chord is greater than the area under the curve and A(a, b) < (1/b +1/a).(b −a)/2 = (a +b).(b −a)/a/b/2 = (b.b −a.a)/a/b/2 = (b/a −a/b)/2, at least for b > a.
Scale the curve in the two directions, to map the curve to itself; since this scales areas by the product of its two scalings of the two co-ordinates, and it's an invariant of the curve precisely when this product is 1, it preserves areas; consequently, A(c.a, c.b) = A(a, b) for all positive a, b, c with b > a. In particular, picking c = 1/a, we obtain A(1, b/a) = A(a, b), so A only depends on the ratio of its two parameters and thus can be re-written in terms of a function of that ratio, which I'll call ln = A(1) = (: A(1, x) ←x :), for which A(a, b) = ln(b/a). The bounds on A now give us 1 −a/b < ln(b/a) < (b/a −a/b)/2, for b > a, whence 1 −1/x < ln(x) < (x −1/x)/2, at least for x > 1.
As noted above, A(a, b) +A(b, c) = A(a, c) whenever 0 < a < b < c; whence ln(b/a) +ln(c/b) = ln(c/a), whence ln(x) +ln(y) = ln(x.y), just as for a homomorphism from addition to multiplication. Thus far, ln is only defined for inputs > 1; we can now extend its definition to positive inputs ≤ 1 by additive completion in the outputs, giving ({reals}: ln :{positives}) with:
This extends ln as a homomorphism from positive multiplication to real addition, making it logarithmic ({reals}: |{positives}). As it is non-zero, the reasoning above tells us it is log(e) for some positive e. By construction, A(a, b) increases monotonically as b increases or a decreases; whence ln(x) increases with x for x > 1. Since 1/x and ln(1/x) = −ln(x) decrease with increasing x, ln(1/x) in fact increases as 1/x increases, for x > 1; whence ln(x) in fact increases monotonically with x for all positive x.
Furthermore, since ln(b) −ln(a) = ln(b/a) and, for b > a, this lies between (b−a)/b and (b−a)/a, we have 1/b < (ln(b) −ln(a))/(b −a) < 1/a. Thus, for any closed interval (1-simplex) about an input x, the range of gradients of chords of ln within that interval lies within the interval between the inverses of its bounding inputs; which may be made as narrow as we like, though it always includes 1/x. Any other gradient differs from 1/x, so we can find a value between it and 1/x whose multiplicative inverse we can use as one end of an interval about x, within which no chord has this other gradient. Consequently, the intersection of gradient-intervals – that contain all gradients of chords of ln within intervals about x – is simply {1/x} and ln is differentiable at x with ln'(x) = 1/x. In particular, as it is everywhere differentiable, ln varies continuously; indeed, it is even smooth.
Now, as ln is logarithmic and non-zero, log(x, y) = ln(y)/ln(x) for all positive x ≠ 1 and y. As ln is monotonically increasing, ln(x) is positive precisely if x > 1 and negative precisely if x < 1; so log(x, y) is positive if x and y are both < 1 or both > 1; while log(x, y) is negative if one of x, y is < 1 and the other is > 1.
As ln is a strictly monotonically increasing homomorphism ({reals}| ln
|{positives}) from positive multiplication to real addition, its reverse is also
a strictly monotonically increasing mapping and a homomorhpism ({positives}|
|{reals}) from real addition to positive multiplication; i.e. an increasing
exponential. We shall, indeed, take it as the
definitive exponential function, give it the
name exp
and express all
other exponentials in terms of it.
Written by Eddy.