# Geometric construction of logarithms

Can you draw a logarithmic scale just using some clever geometric construction? Or can it only be done using an actual table of logarithms?

(It's obviously trivial to draw a linear scale. It isn't hard to draw a scale where the spaces between tick marks doubles at each step. But I can't think of a way to get a logarithmic scale.)

I'm not especially worried about exactly which operations are permitted. I'm really just interested in whether you can make a slide rule without doing a bunch of pencil and paper calculations first...

-
You definitely can't restrict yourself to straightedge-compass... – J. M. May 1 '12 at 12:53
The usual arguments show that ruler+compass points can only be solutions to certain equations, none of which are logarithms – Xodarap May 1 '12 at 12:59
I'm not completely sure, but I gather that for "most values", the logarithm of that value will be irrational (indeed, transcendental). Does that mean that such lengths are "difficult" to construct? – MathematicalOrchid May 1 '12 at 13:02
Straightedge-compass limits you to things that can be expressed in terms of (possibly nested) square roots; neusis lets you do cubics. – J. M. May 1 '12 at 13:17
Usually when I make a slide rule by hand I do it by calculating rational approximations to the logarithms. For example, $2^{10}\approx10^3$, so $\log_{2} 10\approx {10\over 3}$, and then I put a mark at $10\over 3$ and label it 10. The 10 mark should really go at 3.322, not at 3.333, but I cannot mark a piece of paper that accurately anyway. To make a slide rule that can calculate to an accuracy of three decimal places, you only need to calculate the logarithms to three decimal places. – MJD May 1 '12 at 14:08