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Coscott comments on Open thread, Oct. 27 - Nov. 2, 2014 - Less Wrong
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if you memorize logs, I recommend memorizing natural logs of primes. This is all you need to quickly calculate natural log, log2, and log10 of any integer.
You get ln of any number by adding together the natural logs of the prime factors, and you get log_m of n by the formula
log_m(n)=ln(n)/ln(m)
(maybe memorize ln(10) too to make the calculation a little easier)
I can't do real division in my head, but if I wanted to maximise my logarithm-ability while minimizing my number of cards, I would go for logs base (probably 10) of primes, and 1/log(e) and 1/log(2).
But I'm not too fussed about minimizing cards, or about natural logs. Learning more primes might be helpful, but I can get them approximately. E.g. I don't have log_10(11) memorized, but I know it's between log_10(10) and log_10(2*6) which are 1 and 1.08, and it would be closer to the latter (my calculator says 1.041, which is slightly lower than I would have guessed, but if I put it in Anki I'd only go to 1.04 anyway).