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This question already has an answer here:

I was just checking the log table and then suddenly It came to my mind that how were log tables made before calculators/computers how did john Napier calculate so precisely the values of $\log(2),\log(5),\log(7)$ etc .

Is it even possible as I can't even estimate how much time it will take me to do this!! Did Napier use any sort of trick??

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marked as duplicate by Rahul, Henning Makholm, Community Oct 13 '15 at 17:42

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    $\begingroup$ He got an axe, chopped down a tree, sawed some boards... oh. You don't mean that kind of log table. $\endgroup$ – abnry Oct 13 '15 at 17:26
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    $\begingroup$ It took him 20 years! There's some info on wikipedia en.wikipedia.org/wiki/Logarithm#From_Napier_to_Euler $\endgroup$ – Tim Oct 13 '15 at 17:30
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    $\begingroup$ I had the same question a while back $\endgroup$ – Omnomnomnom Oct 13 '15 at 17:31
  • $\begingroup$ @nayrb oh!!Thnx for the update because I was thinking maybe he had made a time machine to come to this century to look for the greatest wood cutter of all time you that is!!! So he could have dinner comfortably!! $\endgroup$ – Freelancer Oct 13 '15 at 17:36
  • $\begingroup$ Thnx @omnomnomnomn got good information from your question mine is repeated I think in a way will delete it $\endgroup$ – Freelancer Oct 13 '15 at 17:40
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Humerous jokes aside about logs, what he did is if I recall correctly was that he worked with the base $1-10^{-7}$ and then computed it's various values at increasing values for numbers between $0$ and $1$. He then used the identity $$\log_a x = \frac{\log_b x}{\log_b a}$$ combined with other logarithmic identities to ease his computation, and I think he choose that value for base for simplicity reasons, which I cannot recall.

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    $\begingroup$ Exactly, cf. e.g. here. $\endgroup$ – Hagen von Eitzen Oct 13 '15 at 17:31
  • $\begingroup$ Fantastic, my memory hasn't failed me. $\endgroup$ – Zelos Malum Oct 13 '15 at 17:32
  • $\begingroup$ See also here. $\endgroup$ – ShreevatsaR May 5 '17 at 9:42
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Lots and lots of paper, pencils and patience. It is not known exactly how Napier produced his table, but it did take him twenty years to do so.

Napier's logarithms did not exactly use modern conventions (and didn't even employ base 10, making it impossible to reuse the same table for different decades), so a generation later Henry Biggs computed the first base 10 logartithm table. He started by taking 54 successive square roots of 10, working to 30 decimal places, until he found the number whose base-10 logarithm is $1/2^{54}$. Together with all the intermediate results this enabled him to raise 10 to various other fractional powers and create a logarithm table. This took another 13 years.

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  • $\begingroup$ And then Euler and company found out a much quicker method. $\endgroup$ – Akiva Weinberger Oct 13 '15 at 17:40

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