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B How to compute logarithm with pen and paper?

  1. Jan 30, 2017 #1
    If you had something particularly nasty like

    ##log_{10}(9)## = ???

    I asked my teacher about how to begin approaching that kind of computation of logarithm.

    He was not very interesyed in explaining the procedures of estimating the value of the log. Perhaps the procedure went beyond the scope of our math course at the time.

    For sure, I think John Napier did not have a calculator which he could just push some buttons. Evidently Napier used some method to compute values which he made into logarithm tables so the logarithm tables could be used by other people to perform calculations.
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  3. Jan 30, 2017 #2


    Staff: Mentor

    Here's some history of how Napier did it:


    Some folks use the series approximation to get a few logarithms and then add, subtract, and multiply them to get the logs of other numbers. As an example, given the log(2) and the log(3) then log(2)+log(3)=log(6) ...

    You could use the appropriate natural log series as the starting point for computing the first natural logarithm and then go from there.

  4. Jan 30, 2017 #3

    Thanks for pointing me in a direction, I will investigate the matter further from there, I think.:woot:
  5. Jan 30, 2017 #4


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    He made the logarithm tables so others didn't have to re-calculate it every time. A lot of Taylor series. Calculators are not magic - they only perform operations every human can do as well. They are just faster.

    For a rough approximation:
    ##\log_{10}(9) = \frac{ln(10\cdot 0.9)}{ln(10)} = \frac{ln(10) + ln(0.9)}{ln(10)} = 1 + \frac{ln(0.9)}{ln(10)} \approx 1+ \frac{-0.1}{2.3} = 1-0.04 = 0.96##.

    Close to the actual value of about 0.9542.
  6. Jan 30, 2017 #5


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    With respect to the natural log, you may consider using a Pade approximation -- this should lead to a much tighter approximation for a given number of terms in a taylor polynomial. The underlying continued fraction is quite intuitive -- though if you are looking for analytical properties, a linear combination of polynomials (i.e. a taylor polynomial) is a lot easier to use than nested fractions.
  7. Jan 30, 2017 #6


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    Mathematicians of old would often hire "idiot savants" who could do amazing math calculations in their heads. They could generate tables. They can not describe how they do the calculations. Some talk about how different numbers taste, feel, or smell.
  8. Jan 30, 2017 #7


    Staff: Mentor

    Do you have a reference for that? I'd be interested in reading about it.

    I don't doubt its true since many great artists actually ran workshops with assistants doing much of the mundane work so perhaps mathematicians used their students in the same way.
  9. Jan 31, 2017 #8
    We have only had the basics of sequences and series taught to us so far.

    Are there any online resources for reading about how series and how they work? I sometimes browse around math stackexchange and saw that series can converge for example.

    I suspect that these are taught somewhere in calculus based courses/books?
  10. Jan 31, 2017 #9


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    No. Calculating numbers is in the realm of numerical mathematics, a very different discipline. By the way, I Googled "approximating logarithms" and got 20 million hits.
  11. Jan 31, 2017 #10


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    I think I read that they used savants in Bell's "Men of Mathematics" long ago (30 years?). The part about the savants trying to describe their methods in terms of taste, smell, feel, etc. was something I gathered from several TV reports over a period of time.
  12. Jan 31, 2017 #11


    Staff: Mentor

    Yes, I've seen those documentaries. There's one guy, Daniel Tammet who was able to recite PI to thousands of places and who was able to be fluent in Icelandic in a very short time. He also apparently has synesthesia where he sees numbers as colors


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