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(Note: limited to $\mathbb{R}$)

(Note: Geometric here means with straightedge and compass)

Standard approaches to introducing the concept of Logarithm rely on a previous exposition of the exponential or simply on that of a power. It then receives the dull definition of "the inverse of the power".

A more intuitive and accessible introduction, which allows doing so even at grade 9, is that of the integer (discrete) logarithm ($\mbox{i}\hspace{-0.15em}\log_b(x)\equiv \lfloor\log_b(x)\rfloor$), i.e., through repeated division by the base without ever getting a result smaller than 1. Ex: 8 can consecutively be divided 3 times by 2 (8/2/2/2) before the results gets smaller than 1. Hence $\operatorname{ilog}_2(8)=3$.

All the usual properties of logarithms can be derived from such a definition, albeit presumably only for integers.

I'm looking, however, for a geometric description for $\log_b(x)$ (not just the integer one $\mbox{i}\hspace{-0.15em}\log_b(x)$) and geometric construction of the integer logarithm $\operatorname{ilog}_b(x)$.

I think I have such a geometric description:

$\log_b(x)$ is the ratio in which a 1/x contraction stretches to 1 relative to the case of a 1/b contraction.

or alternatively

$\log_b(x)$ is the ratio in which a stretch by a factor of x contracts to 1 relative to the case of a stretch by a factor of b.

Example: A contraction of 1/16 can be dilated ("zoom in") 4 times by a factor of 2 to recover the original size, while that of 1/8 can be stretch 3 times by the same factor. Hence $\log_8(16)=4/3$.

By the same definition it is $\log_y(x)\,=\,1/\log_x(y)$ and thus $\log_{16}(8)=3/4$.

The fundamental law of the logarithm should come equally simple from there:

$$\log_b(x)\,=\,\log_{b'}(x)\,\log_b(b')$$

This description of the logarithm is reminiscent of that of the cross-ratio, namely a ratio of ratios, and applies to lengths, areas and volumes.

However, in terms of geometric constructibility, the Gelfond-Schneider theorem would seem to rule that out in most of the cases as $\log_b(x)$ is either rational or transcendental. But what about the integer logarithm,

  1. Is there a construction by compass and ruler of $\mbox{i}\hspace{-0.15em}\log_b(x)$?

  2. If a geometric construction would be impossible, what is the proof or a sketch of it?

  3. Given the above definition in terms of ratio of dilations, would projective geometry provide for a better insight?

  4. In this sense, is that resemblance to the cross-ratio more than a coincidence?

Note: Calculus would seem to provide us with what looks like a geometric description as the area, $A(x)$, of $f(x)=1/x$ between $1$ and $x$. I don't like such an answer, however, because (1) it doesn't provide intuition on how to calculate it (makes an ad hoc reference to an hyperbola) and (2) $\log_b(x)$ is still but the ratio of two numbers, namely, $A(x)/A(b)$, so the above geometric description would seem to encompass this other one.

EDIT: Added pictures:

Turning the dial A of a machine by 1 notch left/right scales the area of all your objects down/up by a factor of two; similarly, dial B works by a factor of 8. Hence, 3 turns of dial A transforms the green, unit square into the orange rectangle, while 4 turns makes it into the big salmon square. How much do you need to turn dial B in order to get the green, unit square into the salmon big square? Ans: 4/3 of a notch.

Turning the dial A of a machine by 1 notch left/right scales the area of all your objects down/up by a factor of two; similarly, dial B works by a factor of 8. Hence, 3 turns of dial A transforms the green, unit square into the orange rectangle, while 4 turns makes it into the big salmon square. How much do you need to turn dial B in order to get the green, unit square into the salmon big square? Ans: 4/3 of a notch.

Is the log an intrinsic projective measure? Poincare hyperbolic distance d_h(p,q) is a projective measure involving the log of a cross-ratio.

Is the log an intrinsic projective measure? FWIW, Poincare hyperbolic distance d_h(p,q) is a projective measure involving the log of a cross-ratio.

As mentioned in my second comment to this post, the discussion in here and the article referenced there may give a hint on this last point. Roughly, a metric tensor on the upper half-plane is given by $ds^2=(dx^2+dy^2)/y^2$, which translates into $ds^2=(dx^2+dy^2)/(1-r^2),\;r^2=x^2+y^2<1$ for the Poincare disc.

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    $\begingroup$ Quick formatting suggestion: $\operatorname{ilog}$ for "$\operatorname{ilog}$" looks a little better than "$i\hspace{-0.15em}\log$". $\endgroup$ – Blue Feb 9 at 8:47
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    $\begingroup$ @Blue Thanks for the suggestion; didn't know about that. I modified half of them by adding a \mbox and the other half as you suggested for comparison. $\endgroup$ – MASL Feb 9 at 9:01
  • $\begingroup$ Maybe a toy problem from physics could help illustrate the relation to a distance, as shown in this question For further details on the optics problem discussed see the paper referenced there $\endgroup$ – MASL Feb 18 at 3:02
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    $\begingroup$ Related : quadrivium.info/MathInt/Notes/DescartesLog.pdf $\endgroup$ – Jean Marie Feb 19 at 21:35
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    $\begingroup$ See also pages 446 and 450 of the very interesting following document by an historian of science about Huygens work on logarithm "function" (anachronism) : hal.archives-ouvertes.fr/tel-01098493/document $\endgroup$ – Jean Marie Feb 19 at 21:51
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Not an answer "per se" but two illuminating references (see below) about 17th century questionning on the connection between geometry and analysis.

These documents address transitions in what could be called the "status" of logarithms. Considered as practical mysterious "numbers" (the vision given by their discoverer, Napier, early 17th century), this status has only gradually moved into our modern vision of "log" as a "function". One of the key discoveries has been that these "logarithms" could be defined as areas under hyperbola $y=1/x$ ; many interesting "disputatios" about the mechanical / non-mechanical nature of this curve, the constructibility of certain of its "values" (the subject of this question) etc. have taken place at that time.

All these debates have allowed to build the modern vision of "log" , established on solid grounds by Euler (mid-18th century)

a) A document about "Descartes Logarithm machine" that can be found there :

http://www.quadrivium.info/MathInt/MathIntentions.html

b) A whole book available online :

"Impossibility results : from Geometry to Analysis"

https://hal.archives-ouvertes.fr/tel-01098493/document

by Davide Crippa, an historian of science. A very interesting document, devoting in particular a large place to the huge work and genial findings of Huygens, that has paved the way to the discovery of calculus (about logarithms, see in particular pages 446-450).

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This description of the logarithm is reminiscent of that of the cross-ratio, namely a ratio of ratios

I'm not sure I'd call this a ratio of ratios. More like the ratio of logarithms of ratios. When you write things like

A contraction of 1/16 can be dilated ("zoom in") 4 times by a factor of 2 to recover the original size

then in my eyes you are taking operation that is in general considered multiplicative, namely that factor 2 expansion, and you shift nomenclature to reat it as additive (in the very general sense of one thing and then another thing) so that your multiplicative wording of “4 times” now describes something that would be exponential in most formalisms. In a way, you have taken the logarithm in your language constructs.

Of course, one thing you can do (and which some instructors have done) is use the good old slide rule to provide a somewhat geometric interpretation of the thing you did there linguistically. The log scale on the slide rule is something that allows you performing multiplications using the mechanics of additions (namely adding lengths). The logarithm (no matter the base) transfers from multiplicative domain to additive domain, and other properties can be derived from that. I'm not sure I'd consider the existence and exact workings of a sled rule as any less artificial than that ad-hoc hyperbola, though, so this may be of limited value.

Another observation involving cross ratios and logarithms: You can get some interesting theorems by multiplying certain cross ratios, in such a way that some factors cancel. You can take the logarithms of cross ratios and obtain things that become interesting when added together. One very surprising connection here is the Laguerre formula, which computes the angle between two lines as the logarithm of the cross ratio of four points at infininity, two of them the ideal circle points which have complex coordinates. Certainly way above 9th grade, needs $\mathbb C$ not just $\mathbb R$, so likely not useful, but I find it interesting.

  1. Is there a construction by compass and ruler of $\operatorname{ilog}_b(x)$?

You can construct $x\cdot y$ following von Staudt: draw an auxiliary line $h$ through $0$. Draw $\triangle(1,P,x)$ where $P$ is an arbitrary point on $h$. Then construct $\triangle(y,P',xy)$ similar to the first triangle, using parallel edges and a point $P'$ on $h$ again. Doing so repeatedly you can construct $b^k$. Obviously $\operatorname{ilog}_b(x)$ (assuming $x>1$) is the largest $k$ such that $b^k\le x$. With a slight change to the original construction you can easily use it for repeated division as well, if you prefer. So if you allow for iteration and for a comparison, then $\operatorname{ilog}$ is constructible. Not sure if this fits your idea of a compass and ruler construction. In a projective setup you can do this even without a compass, using the line at infinity to draw parallels.

  1. If a geometric construction would be impossible, what is the proof or a sketch of it?

I haven't done this, but if you want to show that there can be no way of computing $\operatorname{ilog}(x)$ with a limit on the number of construction steps which does not depend on $x$, then I guess I'd look into results from numeric algorithms used to compute logarithms. My guess would be that they have complexity which depends on the input and on the number of decimals. In a way, $\operatorname{ilog}$ is like asking for zero decimals, but if the input dependence is there in top-of-the-line algorithms, that at least makes a strong case for us not knowing a fixed-length geometric solution. Of course, not knowing a better solution doesn't say there is none, but perhaps some research papers have established absolute lower bounds on the complexity. Haven't checked.

  1. Given the above definition in terms of ratio of dilations, would projective geometry provide for a better insight?

I can't think of anything obvious, and I have some background in projective geometry. I'll try to keep it in mind, will write an update if I manage to think of something better than Laguerre's formula (described above).

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  • $\begingroup$ 0) You seem to suggest this is just a wording thing, with which I can't really agree, except for in a trivial way (as in "a division is a repeated subtraction" is a different wording, sure, but it entails a different view as well). The geometric intuition of this view is, however, clear -at least to me: Take a picture, do 2 different zooming of it. How can you compare them? One way is via the log of one relative to the other. 1) The usual geometric constructions a finite process. That's also what I had in mind. $\endgroup$ – MASL Feb 12 at 18:21
  • $\begingroup$ 2) This whole question of mine gravitates around a geometric construction & interpretation involving a finite number of steps. Writing an algorithm is straightforward when considering that log(x) is but the number of digits of x - 1 (consider base 2). Fixing the number of digits of the result, the complexity is linear on the number of digits of x. My hope in asking point 2) was that maybe this is like asking the quadrature of the circle. It turns out it's impossible because $\pi$ is transcendental. In general, so is log_b(x). But not for say $\log_2(2^n)$. Is this constructible? $\endgroup$ – MASL Feb 12 at 18:29
  • $\begingroup$ 3) The cross-ratio is a ratio of ratios. Its logarithm happens to give rise to a (projective) metric, o better, a metric for the poincare disc model: the hyperbolic distance between two points. On the other hand, as that example of mine shows, the $\log_{8}(16)$, thought geometrically as a comparison of dilations, can be understood as well as a ratio of ratios: 1/16 (the Id transformation rel. to a contraction by 16) "modded" by 1/8 (idem by 8), which gives 4/3. The fundamental law reinforces this idea for me. Going out on limb then, I was wondering if the idea of a projective metric might $\endgroup$ – MASL Feb 12 at 18:45
  • $\begingroup$ (cont.) lead to a geometric characterization, and possibly constraction, of the logarithm. $\endgroup$ – MASL Feb 12 at 18:46
  • $\begingroup$ However, in support to your comment on point 0), you'd say I consider the standard notation somewhat too asymmetric wrt its two arguments. So, instead of $\log_8(16)$ I might prefer something like $|\frac{16}{8}$ that would make the fundamental law of logarithms and that of its "reverse" log (i.e. $\log_{16}(8)=1/\log_8(16)$) more intuitive. $\endgroup$ – MASL Feb 12 at 19:23

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