I am interested in answers to the title question without parentheses, but I found this method below rather interesting, and I am hoping to find it published somewhere, along with a teacher's guide with suggestions on how to present it. (My current target audience consists of U.S. students aged 10 and younger.)
If one wants just powers of 2 (or of some number b), one just takes some ruled filler paper (I like college rules with the lines spaces something like 1/3 inch apart), and on (say) the left edge of the paper places 1,2,4, etc on each rule, and copies this labeling to the right edge of a different sheet. One gets a primitive slide rule to multiply powers of 2 (or of b). A logarithmic scale is just several of these power scales aligned and intertwined, and one needs a good value of (say) (log b / log 2) to get the intertwining right.
Here is the idea. Take a second sheet of ruled paper and (using b=3 as an example) label the rules somewhere 3^0(=1),3,9,27,... successively. Now tilt the paper so that the rule labeled 3^0 intersects the rule on the first sheet labeled 1, with the point of intersection on the edge of the first paper. Tilt the paper and adjust so that the first rule stays fixed at the intersection, but now the second rule labeled 3 intersects the edge between (the rules labeled) 2 and 4. Continue tweaking to get (the intersection points with the edge of the rules labelled) 9, 27, 81, and so on properly positioned (between the rules labeled with powers of 2). When the last tweak is done, mark the edge of the first paper with ticks at the points where the rules labeled 3^i intersect the edge. Now you have the start of a log scale that includes powers of 2 and 3.
One can also add powers of 5 and 7 in an analogous fashion. When one starts with about 28 powers of 2 and fills in powers of 3, 5, and 7 in this fashion, one can then transfer marks to the edge of the second ruled paper and start using this to compute marks for non prime powers, most notably 6 and 10. Compute and add marks as desired.
My search so far reveals some material talking about the basic properties of the log scale and its use as a slide rule. There is likely material on rational approximation of ratios of logs (and this is what is going on with tilting the second sheet, is coming up with a physical measurement of such ratios). However, I have not seen any thing which starts with two sheets of nicely ruled paper and constructs a log scale or a slide rule. Further, the above method does not require exact computation of powers: knowing that 2^19 is near 520,000 and that 3^12 is near 530,000 suffices for the precision expected with this method and tools.
I think this could be useful in multiplication practice as well as an introduction to exponentiation and its inverse. Alternately, it could serve as an example for those wanting to learn more about nomographs and other physical calculating devices. However, it may be that I can't find it because it may be considered too challenging for my intended audience. So in addition to references, I also ask others here what background/age range/maturity level would be wanted in order to show this.
Gerhard "Ask Me About Course Design" Paseman, 2014.12.04
