Comments on: Approximation relating lg, ln, and log10

By: Rational approximations to e — The Endeavour

Rational approximations to e — The Endeavour — Wed, 30 Jan 2013 16:50:06 +0000

[...] The silver ratio Approximation relating lg, ln, and log10 [...]

By: Visto nel Web – 55 « Ok, panico

Visto nel Web – 55 « Ok, panico — Sun, 02 Dec 2012 09:06:30 +0000

[...] Approximation relating lg, ln, and log10 probabilmente OT, ma mi affascinano queste cose; rimpianto di non aver fatto abbastanza mate da piccolo ::: The Endeavour [...]

By: Dave Tate

Dave Tate — Tue, 27 Nov 2012 00:38:10 +0000

Kim, I’m using the fact that logA(x) – logB(x) = ln(x)[ 1/ln(A) - 1/ln(B)]. If you want to approximate ln(x) using a difference of logs of other bases, you want to find integers A, B, and N such that N*[1/ln(A) - 1/ln(B)] is as close to 1 as possible. In the example above, 1/ln(15) – 1/ln(41) is very close to 0.1. Those values of A and B were found empirically.

By: Kim

Kim — Tue, 27 Nov 2012 00:22:17 +0000

Dave, does that integers come from a fractional approximation of e?

By: Dave Tate

Dave Tate — Mon, 26 Nov 2012 22:20:13 +0000

OK, I shouldn’t say “no other approximations of this kind” — you could also do (e.g.)
ln(x) ~ 10 * [log15(x) - log41(x)], which is only off by a factor of 0.000131 …

But that’s not nearly as cool.

By: Dave Tate

Dave Tate — Mon, 26 Nov 2012 22:10:29 +0000

(I seem to be having trouble posting this reply; I suspect my attempts to type inequalities are being misconstrued as really bad HTML…)

Even more cool (to me) is the fact that there are no other approximations of this kind — since 2 is the only integer for which ln(2) is less than 1, it’s the only one that can generate a difference of ~ 1 between 1/ln(r) and 1/ln(k) for integers r and k.

By: Edward Kmett

Edward Kmett — Sun, 25 Nov 2012 19:19:07 +0000

You can of course turn this around to approximate natural log, which is much harder to compute with the two logs every computer scientist already knows how to do in their head.

ln x ~ lg x – log x

e.g.

ln 1024 ~ lg 1024 – log 1024 ~10 – 3 ~ 7 — is close to the actual 6.931471805599453

By: floh

floh — Sat, 24 Nov 2012 22:02:14 +0000

Oki, I didn’t know that with the exception in computer science – learned something new again.
Thanks so far and best regards from Germany

PS.: Great site with a lot of amazing math stuff. Please keep it up!

By: John

John — Sat, 24 Nov 2012 21:22:06 +0000

I'm quoting the formula as I found it in Knuth's book. He uses lg for log base 2, which is fairly common in computer science, at least in the US. More explicitly, log₂ x ≈ log_e x + log₁₀ x.

By: floh

floh — Sat, 24 Nov 2012 21:15:00 +0000

Maybe a mistake that causes confusion!?:
“lg” is the short form for “log base 1o” and not “base 2″.
But afaik “log base 2″ has the short form “ld x” or “lb x”!?