19
$\begingroup$

I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits. By pen and paper that is. I'm doing this old school.

What first came to mind was to use $\log(ab) = \log(a) + \log(b)$ for reduction.
And then use the taylor series for $\log(1-x)$ when $-1 < x \leq 1$ But convergence is rather slow on this one.

Can you come up with a better method?

  • 0
    Look at [this thread here](http://math.stackexchange.com/questions/47927/motivation-for-napiers-logarithms) for some tricks that reduce the work and relevant background information. Essentially you don't want to use a single power series but do most of the work by using appropriate interpolations.2011-09-01
  • 0
    The Feynman Lectures on Physics Vol1 Ch22 has the same ethos as the question and some good insight too.2011-09-01
  • 1
    You can find some actual calculations in this column: http://maa.org/editorial/euler/How%20Euler%20Did%20It%2021%20logs%20.pdf (it has an iterative method for base-10 logs, and half-way down page 3 it uses (a friendlier version of) the method in @Iasafro's post). Note that you need only use the series for reciprocals of integers since you can piece everything else together from those (and some ingenuity on your part). I also want to add that it's nice seeing someone else who is interested in this stuff (whenever I feel my arithmetic skills are declining I add to my log tables for practice).2011-09-02
  • 3
    You are seriously doing this for fun? I know that my parents had to use logarithm tables when there were no affordable calculators. Why would you want to do this by hand today?2011-09-02
  • 2
    @Raphael: While it's true that calculating things by hand is not that efficient when you want _results_, it is good _practice_ at arithmetic. From my personal experience, as well as observing others, it seems that the jokes about basic skills (arithmetic, trig, calculus, etc.) degrading the higher one goes in mathematics are all too true. To counteract this, whenever I notice I'm getting sloppy with numbers I just pull out my table and add to it :) A slight digression: Another use for calculating things by hand (well, maybe not always _by hand_) is to help calculus students understand what..2011-09-03
  • 0
    ...convergence really means. Saying "$\sum_{n=0}^{\infty} \frac{1}{n!}$ converges by the ___ test" is one thing; showing them how fast zeroes get inserted after the decimal point of $\frac{1}{n!}$ for increasing $n$ is another. I'm not sure about Eberhard, but those are the two reasons I give people when I mention that I have my own log (and trig) tables.2011-09-03
  • 0
    I am not sure that calculating anything (besides really basic stuff, maybe up to dealing with rationals) by hand is of use nowadays. It seems to me that the better exercise would be to sit down and learn how to quickly write a small programm that does the computation, or learn how to use `bc` etc. As a computer scientist, I am probably biased here.2011-09-03
  • 1
    @Raphael As a computer scientist, aren't you interested in what method would do this quickest? Whether it's a human or computer computing? Robert's first solution takes about $2250$ multiplications (not bad). Iasafro's solution takes about 900 divisions and 450 cubings, and 450 raisings to 5. Robert's second solution takes about 2700 divisions, 900 squarings, and 900 cubings. Mine takes 1700 divisions. These kinds of comparisons should be important for making an efficient algorithm. Even if its just so the ideas can be applied to "bigger" problems.2011-09-30
  • 0
    That's right, but the OP explicitly stated he wanted to create a table by hand. I do not oppose finding efficient procedures, on the contrary, but I question the need for a table and the accuracy of doing long calculations by hand.2011-10-01
  • 0
    I very nearly needed to do this today when I forgot my calculator for a chemistry test, and was unable to borrow one. Fortunately the problems were all overcooked and I didn't need to use anything past long division, but you never know...2015-10-23

6 Answers 6