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From: David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz>
Newsgroups: rec.puzzles
Subject: Re: Log (base 2) of 3   --    (without a Calculator)
Date: Sat, 30 Nov 2024 09:18:33 -0000 (UTC)
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On Tue, 19 Nov 2024 19:47:00 +0000 (UTC), Richard Tobin wrote:

> In article <97e7e6fb078310c8d4d600c247847957@www.novabbs.com>,
> HenHanna  <HenHanna@dev.null> wrote:
> 
>> i wonder if there's a way to get better (and better) approximations.
> 
> Look for more powers of 2 near to powers of 3.
> 
> For example,
> 
> 3^7  (= 2187)  > 2^11 (= 2048),  so 3 > 2^(11/7),  so log2(3) > 11/7 =
> 1.571+
> 3^10 (= 59049) < 2^16 (= 65536), so 3 < 2^(16/10), so log2(3) < 10/6 =
> 1.6
> 
> 3^12 is very close to 2^19, so log2(3) is very close to 19/12 = 1.583+
> 
> -- Richard

That's a very nice explanation.

Not a direct answer to HenHanna's original question, but interesting none 
the less. I was trying to work out how Babbage's difference engine, using 
finite differences, could be used to perform relaterd calculations.I got a 
bit distracted, but the following translation of Briggs' ARITHMETICA 
LOGARITHMICA was very informative.

https://www.17centurymaths.com/contents/albriggs.html 



-- 
David Entwistle