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Most units, e.g. for length, time, mass, etc, are linear in relation to the underlying property.

But there are some special quantities, where the units can vary several orders of magnitude within the same system, so they are measured using logarithmic scales. Such cases are for example electromagnetic signal, where we have dbm, or earthquake intensity (Richter scale), or sound intensity (db).

My question is are there any physical quantities that vary so much within the same system, where the units are expressed in log(log) scale? Something that could change orders of magnitudes of orders of magnitudes?

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If you decide to measure the distance of the farthest galaxy ( 13.3 billion light years away) in nanometers u will get something like 10^35 Log of that is 35 and Log log of that is 1.544

Seems pretty silly doesn't it?

The actual problem is why we use log scale. It is bacuase one parameter varies by several orders of magnitude when another changes linearly. .

Occasionally you use log(a) vs log (b) plots. The accuracy however becomes extremely poor in some sections. A log {log (a)} vs b variation is useless.

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  • $\begingroup$ Not my downvote on your answer, but I guessing it's because the example you give is about length, and it shows that loglog is not appropriate for that measure. Indeed, for length I am not aware of even a logarithmic scale. But for other quantifies, logarithmic is clearly needed. The point of this question was to see if any such quantity exists, where people have ever needed log(log()) for anything practical. $\endgroup$ – user000001 Dec 31 '18 at 11:14
  • $\begingroup$ I used length to show that even in that example the total length is 35 orders of magnitude. So the log log is simply 1.5. so you are not going to find any physical variable that would coblver a range sufficient for log log. $\endgroup$ – Dr S T Lakshmikumar Dec 31 '18 at 13:02

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