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I was randomly re-watching some Youtube video on solving exponential equations and was poking around solving some, using the log() function in R to do the logs,

I was randomly re-watching some Youtube video on solving exponential equations and was poking around solving some, using the log() function in R to do the logs, and got to thinking about the "old days" when you had to look these things up in log tables. And that led me to think "Wait, how did the first log tables get written in the first place? And for that matter, how exactly do you work out a log value the manual way anyway???" A few moments of Googling turned this up. Thought some other folks might find this topic interesting as well. One fun aspect of all this, is that it forces you to consider that thinking of exponentiation as "repeated multiplication" is a bit dodgy when you allow for decimal exponents. I mean, given 3^2.5, what does it mean to say "multiply 3 by itself 2.5 times?" Actually, thinking about that was the thing that really set me off on an exploration of how the heck they worked out these log values in the beginning.

arbitrandomuser 3y ago
A very good exposition of this topic can be found in the Feynman lectures https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html
movpasd 3y ago
The basic idea of how to generalise "repeated multiplication" to the whole real continuum is to think about multiplication as a kind of scaling. Going from 2^n to 2^(n+1) is ultimately about scaling the result up by a factor of x2. So, 2^(n+0.5) ought to be about scaling up halfway to x2, so that if you were to repeat that operation again, you got x2; and that's precisely sqrt(2). That reasoning gets you all the rationals. Since rationals are dense in the reals, impose continuity and you get all reals. This is the root of the algebraic definition of exponentials. But where there are reals there is calculus, and we also get a lot of insight from a differential definition. The key insight that leads to this definition is that if we could break down the exponential to 2^(n+epsilon) for a very small epsilon, we could stack up O(1/epsilon) of them to get wherever we like on the real continuum. So it makes sense that the definition dy/dx = a.y should produce the same function. This notion of "break down an operation into infinitesimal bits that stack up" can be taken seriously and formalised, and if you do that you end up with the theory of Lie groups and Lie algebras.
jacobolus 3y ago
> 3^2.5, what does it mean to say "multiply 3 by itself 2.5 times?" It means 3 · 3 · √3. If you write the exponent in binary, you can approximate any such power as a product of various repeated square roots. > how the heck they worked out these log values in the beginning If you're really curious about the very beginning, here's an English translation of Briggs's (1624) book: https://17centurymaths.com/contents/albriggs.html https://17centurymaths.com/contents/albriggs.html And here are English translation of Napier's (1614, 1619) books: https://17centurymaths.com/contents/napiercontents.html https://17centurymaths.com/contents/napiercontents.html
- svat 3y ago
  A few years ago, I wrote a summary of Napier's conception and construction here: https://math.stackexchange.com/questions/47927/motivation-for-napiers-logarithms/48036#48036 https://math.stackexchange.com/questions/47927/motivation-fo...
- lifthrasiir 3y ago
  Wikipedia has a concise enough explanation of Napier's invention and its peculiarity compared to the modern definition [1]. [1] https://en.wikipedia.org/wiki/History_of_logarithms#Napier https://en.wikipedia.org/wiki/History_of_logarithms#Napier
  - mindcrime 3y ago
    Of course Wikipedia has a page titled "History of logarithms"! I should have just looked there first. :-)
    - jacobolus 3y ago
      A better source is Cajori's (1913) 7-part history of logarithms in the American Math Monthly, [20(1) 5–14](https://www.jstor.org/stable/2973509 https://www.jstor.org/stable/2973509), [20(2) 35–47](https://www.jstor.org/stable/2974078 https://www.jstor.org/stable/2974078), [20(3) 75–84](https://www.jstor.org/stable/2973441 https://www.jstor.org/stable/2973441), [20(4) 107–117](https://www.jstor.org/stable/2972960 https://www.jstor.org/stable/2972960), [20(5) 148–151](https://www.jstor.org/stable/2972412 https://www.jstor.org/stable/2972412), [20(6) 173–182](https://www.jstor.org/stable/2973069 https://www.jstor.org/stable/2973069), [20(7) 205–210](https://www.jstor.org/stable/2974104 https://www.jstor.org/stable/2974104)

arbitrandomuser 3y ago

A very good exposition of this topic can be found in the Feynman lectures https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html

movpasd 3y ago

The basic idea of how to generalise "repeated multiplication" to the whole real continuum is to think about multiplication as a kind of scaling. Going from 2^n to 2^(n+1) is ultimately about scaling the result up by a factor of x2. So, 2^(n+0.5) ought to be about scaling up halfway to x2, so that if you were to repeat that operation again, you got x2; and that's precisely sqrt(2). That reasoning gets you all the rationals. Since rationals are dense in the reals, impose continuity and you get all reals. This is the root of the algebraic definition of exponentials. But where there are reals there is calculus, and we also get a lot of insight from a differential definition. The key insight that leads to this definition is that if we could break down the exponential to 2^(n+epsilon) for a very small epsilon, we could stack up O(1/epsilon) of them to get wherever we like on the real continuum. So it makes sense that the definition dy/dx = a.y should produce the same function. This notion of "break down an operation into infinitesimal bits that stack up" can be taken seriously and formalised, and if you do that you end up with the theory of Lie groups and Lie algebras.

jacobolus 3y ago

> 3^2.5, what does it mean to say "multiply 3 by itself 2.5 times?" It means 3 · 3 · √3. If you write the exponent in binary, you can approximate any such power as a product of various repeated square roots. > how the heck they worked out these log values in the beginning If you're really curious about the very beginning, here's an English translation of Briggs's (1624) book: https://17centurymaths.com/contents/albriggs.html https://17centurymaths.com/contents/albriggs.html And here are English translation of Napier's (1614, 1619) books: https://17centurymaths.com/contents/napiercontents.html https://17centurymaths.com/contents/napiercontents.html

svat 3y ago
A few years ago, I wrote a summary of Napier's conception and construction here: https://math.stackexchange.com/questions/47927/motivation-for-napiers-logarithms/48036#48036 https://math.stackexchange.com/questions/47927/motivation-fo...
lifthrasiir 3y ago
Wikipedia has a concise enough explanation of Napier's invention and its peculiarity compared to the modern definition [1]. [1] https://en.wikipedia.org/wiki/History_of_logarithms#Napier https://en.wikipedia.org/wiki/History_of_logarithms#Napier
- mindcrime 3y ago
  Of course Wikipedia has a page titled "History of logarithms"! I should have just looked there first. :-)
  - jacobolus 3y ago
    A better source is Cajori's (1913) 7-part history of logarithms in the American Math Monthly, [20(1) 5–14](https://www.jstor.org/stable/2973509 https://www.jstor.org/stable/2973509), [20(2) 35–47](https://www.jstor.org/stable/2974078 https://www.jstor.org/stable/2974078), [20(3) 75–84](https://www.jstor.org/stable/2973441 https://www.jstor.org/stable/2973441), [20(4) 107–117](https://www.jstor.org/stable/2972960 https://www.jstor.org/stable/2972960), [20(5) 148–151](https://www.jstor.org/stable/2972412 https://www.jstor.org/stable/2972412), [20(6) 173–182](https://www.jstor.org/stable/2973069 https://www.jstor.org/stable/2973069), [20(7) 205–210](https://www.jstor.org/stable/2974104 https://www.jstor.org/stable/2974104)