# Scaling ratio: the logarithm

This post is intentionally just a stub.

In a previous post Number 1 I mentioned an intuitive way of thinking on the logarithm. However, it is clearly too cumbersome: sometimes it’s about multiplication, others division, we have to add ad hoc a sign, and the interpretation of the fundamental law of logarithms $\log_b(x)\,\log_{b'}b\,=\,\log_{b'}x$ required us to think of the term $\log_{b'}(b)$ as something slightly different than $\log_b(x)$.

It was good for talking about the integer logarithm, but it clearly isn’t the whole story. I still do not have the right one, but I think I have now a better way to think about the logarithm.

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

Example: A contraction of 1/16 can be dilated 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 $\log_8(16)=4/3$. I think this means we can swap contractions by 1/x by stretches by x, and stretches by contractions in this definition of logarithm.

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:

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

Given the factorization of x and y, $x=p_1^{a_1}\dots p_k^{a_k}$ and $y=q_1^{b_1}\dots q_r^{b_r}$, with $p_i,\,q_j$ prime numbers, we have

$\log_y(x)\,=\,\sum_i^k\,a_i\,\log_y(p_i)\,=\,\sum_i^k\,a_i\,\frac{1}{\log_{p_i}(y)}\,=\,\sum_i^k\frac{a_i}{\sum_j^r\,b_j\,\log_{p_i}(q_j)}$

and the logarithm of any two  numbers is a function that depends on the log of any pair

of prime numbers. What is then something like $\log_2(3),\;\log_2(5),\;\log_3(5),\dots$?

So the logarithm is more about the geometric transformation of dilations, hometheties or scalings. This in turn leads to a projective transformation.

Random thoughts -or not so random, as I intentionally want to take some big steps back and see where this idea could lead to:

Does the $\log$ have a natural geometric interpretation in a projective space? I don’t expect a geometric construction, just an interpretation similar to those provided by the figures in this thread.

Does the $\log$ have a natural topological interpretation as (properties of) homeomorphisms?

Does the $\log$ “carry” a natural generalization to a non-flat manifold leading to something different? For instance, contract/stretch a “line-segment” along a given geodesic on a given manifold. Would this relate to Tarantola’s autoparallel geovectors?

Edit: I realized I didn’t use what seems the usual definition (order) for the cross-ratio. For four points on a line O,p,q,E, in that order, it is $[p,q;O,E]=(O-p)\,(E-q)/((E-p)\,(O-q))$. Ultimately, there are only 6 combinations all related by fractional transformations.

Using this definition consistently, though, we can use the cross-ratio $[p,q;O,\infty]$ to express the factor by which to contract the segment Oq to match Op. See 3rd figure below.

The scaling factors $\frac{AE}{AH}=1/2$ and $\frac{AE}{AK}=1/3$ coincide with the cross-ratios $[A,E,H,\infty]$ and $[A,E,K,\infty]$, respectively. Similarly for the other pairs.  Whence it follows that the Poincare non-euclidean distance $d(E,H)$, or rather a ratio of them, gives us the logarithm of one of the scaling factors relative to the other.

Of course, this would be a circular definition and it doesn’t provide an independent way to calculate the logarithm. For that we would need a way to determine the non-euclidean distances $d(E,H)$.

Using logarithms of “numbers” instead of the numbers themselves is not new. The Logarithmic Number System (LNS) provides al alternative implementation of arithmetic operations that can be faster than using floating point arithmetics. For any number $X$ we assign $X\;\to\;\{s_x,\,x\equiv \log_b(X)\}$.

It’s interesting to see how this relates to the cross-ratio. For the points $0,1,x$ we have $[x,1;0,\infty]=x$ and $\log_b(x) = \log_b [x,1;0,\infty] \equiv d_b(1,b)$, i.e., the non-euclidean Poincaré distance between 1 and $x$.

But most importantly, this makes it clear that the LNS is invariant under projective transformations.

Furthermore, this is equivalent to what Tarantola did for Lie Groups. Would it make sense to ask how LNS would have to be modified if use it for $S^1$? Does the geometry of the circle enters the game as does for the case of Lie groups? Maybe this could be like a trivial example of his approach.

I’m strongly influenced here by Tarantola’s discussions, both his in-depth development as well as shorter essays/notes on the subject. It would seem that engineers, since long ago, have been using the logarithm

as a way to compare relative attenuations, amplifications,.. albeit with some definition involving some odd factors, e.g., the decibel.

He ends that text with the following concluding remark that summarizes his thoughts and this thread:

The original unit is the nonlinear one. The “logarithmic” unit is natural. And linear. Note: try to make obvious that I am making more than using logarithmic scales. I try to give life to the logarithmic parameters.

Note for me: Each $X$ is projected onto $X\,\to\, \arctan (1/X)$

I’m missing something: is there an intrinsic measure of $[D,H;L,M]$ in $S^1$ such that $[D,H;L,M]=[F,G;K,\infty]$? Oops! That last figure is wrong! The CR doesn’t project onto the circle as stated!! ;-<  [1,2]