The largest commercial cylindrical slide rule has a scale length of 24m

You'd be multiplying the square roots, not exponentiating.

Yeah, I don't know how practical a construction would be overall for someone who seriously went at it; I was mainly moved to answer because this objection that constructions can't express transcendental numbers just doesn't seem relevant -- digitally you don't keep infinite precision either.

Square roots and multiplying are two of the simplest geometric constructions (geometric mean and taking a proportion), compass and straightedge operations can be accurate, you can construct it enlarged and then scale it down, and finally the precision you need to aim for at the end is bounded. A potentially fun project idea. Think of it as a kind of retrocomputing: how might Euclid or Archimedes have designed a slide rule?

I have a favorite circular slide rule: the E6B Flight Computor (sic). Pilots still use them to calculate airspeeds, distances, and fuel burn; and the reverse side can be used to calculate course corrections in the face of wind. Actually a lot of pilots probably use the iPad they keep all their air charts on for this, but it's great to have as a backup. All those flight parameters can be found with a simple mechanism that's made of card stock and uses no electricity! Sometimes the most marvelous tech is not high tech.

I remember seeing a small exhibit of slide rules including some cylindrical ones at MIT in the early 70's. It was in a few display cases in one of the older buildings (I think it was building 1, but it could have been building 5--they are connected.) We all had slide rules back then, mine was a bright yellow model made by Pickett and made of aluminum--it never operated as smoothly as the ones made of bamboo by K&E

I bought my slide rule in 7th or 8th grade with money I made from my paper route around 1964; I still have it and it functions just as well as when it was new.

I mentioned that we all had them, this was because there was really no alternative. Slide rules could perform all sorts of calculations, multiplication, division, exponentials, trig functions, all to an accuracy of around 3 significant digits. Computers existed, but terminals (with few exceptions) did not. Any calculation with a real computer required getting to the comp center and punching cards (any mistake on their funny keyboards meant throwing the card away and starting over). After that one would have to submit the deck to an operator behind a glass window. In ten minutes to an hour you would get the first result of running your program produced on fan fold paper, usually with green and white tinted background on the paper. The printing was done with a line-printer, often in upper case only. Naturally, the first few attempts ended up with syntax errors and it was easy to waste a lot of time on simple calculations so the 3 significant digits of an ordinary slide rule started to look good enough.

Of course, we all also had (right next to our collegiate desktop Websters dictionary) a copy of the CRC Standard Mathematical Tables. This book, hundreds of pages long, had tables of logarithms and trig functions accurately to 5 digits which could solve these problems to 5 or 6 digits accuracy with careful interpolation techniques (which were taught in high school back then).

A year or two after I bought my slide rule I saw an early black and white episode of Lost in Space, a 60's TV show about a family lost on a remote planet. On the program the character Will Robinson was using a handheld calculator. It was pure science fiction device about the size of large 6cm thick hardback book. It contained a keyboard of perhaps 16 keys and a large, say 5cm, high display. I thought that it was so incredible, like the space suits and robots that appeared in the show. I wished that there was someway in the distant future that I would ever have such an amazing device!

As a follow up, I wanted to know what clever mechanism was used to generate the divisions on slide rulers... I figured someone, somewhere and a linkage that used the fact that the derivative of ln(x) is 1/x... nope... they used a big cam.

Relevant articles:

https://sliderules.lovett.com/cookiedev/extendeddisplayartic...

https://sliderules.lovett.com/cookiedev/extendeddisplayartic...

Also, not actual a logarithm, but a wicked clever hack:

https://en.wikipedia.org/wiki/Irish_logarithm