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I just read that logarithms were not initially defined in terms of their inverse relationship to exponential functions (and that Euler was the first to develop them in this way).

So how were they initially defined? All I could find on an (admittedly cursory) internet search was that logarithms were used to streamline multiplication and division, and also the calculation of "sines", for which extensive tables were constructed. Does anyone know how these tables were made?

And is there a reason that the etymology of "logarithm" seems to be "ratio-number"? From dictionary.com:

1610s, Mod.L. logarithmus, coined by Scot. mathematician John Napier (1550-1617), lit. "ratio-number," from Gk. logos "proportion, ratio, word" (see logos) + arithmos "number" (see arithmetic). Related: Logarithmic.

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marked as duplicate by MJD, Michael Albanese, user147263, Lucian, Macavity May 27 '14 at 4:14

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  • $\begingroup$ en.wikipedia.org/wiki/Napier%27s_bones but they say the bones were not the same as logarithm, en.wikipedia.org/wiki/Logarithm#From_Napier_to_Euler $\endgroup$ – Will Jagy May 26 '14 at 22:25
  • $\begingroup$ The truly excellent book "e: The Story of a Number" explains this and all sorts of wonderful other things about e and logs. It is clear, and at a level understandable for anybody just starting calculus or beyond. $\endgroup$ – Mark Fischler May 26 '14 at 22:26
  • $\begingroup$ @MarkFischler Thanks, I will look into that. Do you happen to have the book available? A relevant quotation might be nice. $\endgroup$ – user142299 May 26 '14 at 22:32
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    $\begingroup$ This answer explains it in detail $\endgroup$ – MJD May 26 '14 at 22:56
  • $\begingroup$ The greek term for "ratio-numbers" was ritós arithmós, so in this context logos arithmós might better be translated as "word numbers" or "reasoned numbers" because the numerical values of logarithms couldn't be found through mechanical arithmetic all operations. Instead, the logarithm operation had to be explained in words, and their values were found by reasoning through the appropriate tables. $\endgroup$ – David H May 26 '14 at 23:28