Wikipedia:Reference desk/Archives/Mathematics/2020 October 12

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October 12

Logarithm 2

A closely related method can be used to compute the logarithm of integers. Putting $\textstyle z={\frac {n+1}{n}}$ in the above series, it follows that:

$\log(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.$

If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1), with a rate of convergence of ${\frac {1}{2n+1}}$.

This part is unsourced. I'm looking for its source. Can anyone please help? 14.186.0.207 (talk) 17:50, 12 October 2020 (UTC)

This goes back to Euler or even earlier. Euler uses this series in the book Introductio in analysin infinitorum to compute logarithms of integers in the way I've explained here. Count Iblis (talk) 18:22, 12 October 2020 (UTC)

To me the fact that Euler used this idea to compute the log of 2, 3 and 5 is more significant than the series, which is really just a trivial transformation of the series for hyperbolic arctangent. Imo either delete the passage altogether or expand it to include a summary of the SE post. (Not to be use a reliable source btw.) I think it may only be of historical interest now, so maybe also move it into the history section. Euler can be used as a reference but I assume it's in Latin and several hundred years old so that may make the reading rather difficult. An additional source for the historical relevance would be nice as well. --RDBury (talk) 19:08, 12 October 2020 (UTC)

PS. I found a translation here, Euler does use something very similar to the series, but not quite, toward the end of the chapter. He gives a table of log n to 25 decimals for n=2 to 10; pretty impressive considering the electronic computer won't be invented for another few centuries. Presumably you could get log 11 by expanding the series for log(121/120). --RDBury (talk) 19:35, 12 October 2020 (UTC)

Does it matter whether $n$ is an integer?  --Lambiam 21:42, 12 October 2020 (UTC)

For the series, no. Euler only applied it to integers though. The general formula is

$\log a-\log b=\log {\frac {a}{b}}=2\operatorname {arctanh} {\frac {a-b}{a+b}}\ \ {\rm {for}}\ |a-b|<a+b$

where arctanh u is computed by the series

$\operatorname {arctanh} u=u+{\frac {u^{3}}{3}}+{\frac {u^{5}}{5}}+\dots \ .$

The idea is that with a little manipulation you can reduce the computation of two or more logs to the computation of the same number of logs but with the |a-b|≪a+b so the series converge quickly. The SE post does with with log 2, log 3 and log 5, but you can generate similar examples ad nauseam, e.g.:

$\log 2=5\log {\frac {9}{8}}+2\log {\frac {256}{243}}=10\operatorname {arctanh} {\frac {1}{17}}+4\operatorname {arctanh} {\frac {13}{499}}$

$\log 3=8\log {\frac {9}{8}}+3\log {\frac {256}{243}}=16\operatorname {arctanh} {\frac {1}{17}}+6\operatorname {arctanh} {\frac {13}{499}}$

Similarly log 2, log 3, log 5 and log 7 can be written as integer combinations of log(49/48), log(50/49), log(64/63) and log(81/80), starting with

$\log 2=12\log {\frac {49}{48}}+5\log {\frac {50}{49}}+14\log {\frac {64}{63}}+10\log {\frac {81}{80}}.$

But I assume this is mainly of historical interest since presumably the methods laid out in the next two sections (AGM and Feynman) supersede this idea. --RDBury (talk) 09:25, 13 October 2020 (UTC)

Speaking of Feynman, and reading ahead a bit, is anyone able to find a more concrete description of the Feynman algorithm other than the Physics Today article? The article (or a reasonable facsimile) is available here, but it's very vague about how the algorithm would actually work. Other than a few confused SE posts referencing either that article or the WP article I couldn't find anything any more specific. One of the claims in the PT article, that every number between 1 and 2 has a unique representation as a product of factors of the form (1+2^-k) is easy to disprove and is covered in the SE posts. For example:

$3/2=(1+2^{-1})=(1+2^{-2})(1+2^{-3})(1+2^{-4})(1+2^{-8})(1+2^{-16})\dots$

The WP article gives somewhat more detail than is given in the PT article, so it's tempting to give it a {{Failed verification}} tag, but it's an edge case. --RDBury (talk) 12:16, 13 October 2020 (UTC)

@RDBury: doesn’t sound like an edge case to me; rather clear-cut, actually. Certainly worth a talk-page comment! —JBL (talk) 12:05, 14 October 2020 (UTC)

@JBL:: The thing is the method seems like a valid and efficient way of computing logarithms and probably the only way that the brief outline given in the PT article could be fleshed out into a working method. (Some of the wording, however, might imply the actual representation used is a product of factors of the form (1-2^-k)^-1, which seems a bit simpler to me.) PT is, by Wikipedia standards, a reliable source, but it's like trying to figure out quantum mechanics from an article about Brian Cox in People magazine. --RDBury (talk) 02:44, 15 October 2020 (UTC)

PS. I went ahead and added a 'single source' tag. --RDBury (talk) 03:14, 15 October 2020 (UTC)