Geometric interpretation of the Logarithm (in $\mathbb{R}$)

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).

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.

2. 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.

3. 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).