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I'm trying to make a Log or Antilog table small enough to fit in the back of a wallet calendar (or a business card). My intend is to build a mathematically useful gift that can be used by anybody eager to spend 5 minutes learning how it works.

To simplify the question let's assume that I'm able to fit just 50 numbers in the table (with some spare space to add some useful logarithmic formulas and some additional constants like $\log_{10}(2)$, $\log_{10}(e)$ or $\log_{10}(\pi)$).

My question is: What 50 numbers should I select to maximize the usefulness of the table?


My first idea is to select the decimal logarithms of the first 50 even numbers:

 x   | Log(x)
--------------
  2  | log(2)
  4  | log(4)
 ... | ...
 100 | log(100)

But it turns out that the first 5 logarithms are not necessary at all since: log(x) = log(10·x)-1.

I can put them in the table anyway or just use 45 numbers (which are both sub-optimal options from my point of view) or simply re-scale the whole table:

   x   |  Log(x)
------------------
  11.8 | log(11.8)
  13.6 | log(13.6)
  ...  |    ...
 100.0 | log(100.0)

But if I'm not going to be able to obtain the logarithm of most natural numbers maybe it is a better idea to just make an Antilog table:

   x  |   AntiLog(x)
---------------------
 0.00 | antilog(0.00)
 0.02 | antilog(0.02)
  ... |    ...
 0.98 | antilog(0.98)

I know that both kind of tables (logarithmic or antilogarithmic) can be used to compute logarithms and antilogarithms approximately (using a direct or a reverse look-up depending on the case) but I'm not sure if there is a good reason to prefer the former ones instead of the latter ones.

One possible argument in favor of logarithmic tables is that they can be used to compute the logarithm of a very big number as long as this number has small factors since log(a·b) = log(a)+log(b). With an antilogarithmic table you can only approximate the logarithm of a prime number and, therefore, you will only obtain an approximate answer (since the logarithms are irrationals in general, here approximate should be read as with less precision than the numbers in the table offer).

  • Are there more arguments in favor of logarithmic tables (versus antilogarithmic tables)?

  • Are them strong enough to justify the waste of the 10% of the table?

  • Are there some applications where each type of table is preferred over the other?

  • Is one operation used more often than the other in practice?

  • Assuming that both operations are equally likely to be used, and that a direct look-up provide better precision than a reverse look-up, which operation should I privilege? Are the interpolation errors equally severe in both cases?

Related Questions:

  • Since the precision on the left column of the table is just 1/50 (in the sense that we are going to divide either [1,100] or [0, 1) into 50 equidistant points), I'm not sure about how much precision should I use in the right column. Does it make sense to fit as many decimal places as possible or is it just a waste of ink and readability?

  • Can you suggest a better content for a table like this? For example, following a previous argument it may be better to make a table of the logarithms of the first 50 prime numbers (finding a logarithm will be slightly more difficult and that's OK but I'm not sure if that kind table will allow me to find antilogarithms or arbitrary numbers with a reasonable precision). A good trade off may be to list the logarithms of the first 50 odd numbers (instead of the first 50 even numbers) to include all prime numbers smaller than 100.

  • Which formulas and constants would you add to increase the usefulness of the gift? (remember that the space is really limited)

Example:

This is an example of what I mean by Decimal Logarithms Table in a Business Card:

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  • 2
    $\begingroup$ What do you expect a Log or Antilog table to be useful for? $\endgroup$ – lhf Feb 27 '15 at 12:08
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    $\begingroup$ @lhf Log tables have been used historically to carry out some difficult arithmetic operations like finding the Kth root of any given number. I expect this little table to allow me to effortless produce some back-of-the-envelope computations without the help of an electronic calculator. Basically the same applications that a small slide rule had for a layman about 50 years ago. $\endgroup$ – Carlos Luna Feb 27 '15 at 12:37
  • $\begingroup$ The two are equivalent. If you have a table of logarithms, you can use it to find anti-logarithms and if you have a table of anti-logarithms you can use it to find logarithms. $\endgroup$ – user247327 May 24 '18 at 16:54
  • $\begingroup$ @user247327: Please, take the time to read the whole text (not just the header) before answering a question. $\endgroup$ – Carlos Luna May 26 '18 at 14:36
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I really like this question. It probes the meaning of what is "useful" to a mathematician.

First off, I would definitely choose log rather than antilog tables. I presume we are talking about natural logarithms, by the way.

Then comes my list of 50 numbers...

  1. My first choice is 2.

  2. My second choice is 10. I don't need $log_{10} (2)$ because I can just divide the first two.

  3. I would like $\pi$. I also like the fact that it comes third in my list!

  4. I would like $g$ because I am an applied mathematician.

  5. I expect there are a few other constants like $c$ and Avogrado's that would be useful.

  6. I would choose 3.

  7. I don't need to choose 5 because it is $\frac {10} 2$, so I would choose the remaining prime numbers starting with 7.

If instead we are using logarithms to base 10...

  1. My first choice is 2.

  2. My second choice is e. This will allow me to convert to natural logarithms. Maybe log(log(e)) would be useful, too, because I will need to divide by log(e) often.

  3. I would like $\pi$. I also like the fact that it comes third in my list!

  4. I would like $g$ because I am an applied mathematician.

  5. I expect there are a few other constants like $c$ and Avogrado's that would be useful.

  6. I would choose 3.

  7. I don't need to choose 5 because it is $\frac {10} 2$, so I would choose the remaining prime numbers starting with 7.

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  • $\begingroup$ Well, I really expect my table to allow me to compute arbitrary logarithms and antilogarithms without too much effort and, therefore, choosing a base 10 logarithm seems a better choice (you can easily compute the characteristic, look-up the mantissa on the table and add them both). That's why I planned to add $\log_{10}{e}$ and $\log_{10}{2}$ to be able to compute also logarithms in other bases. $\endgroup$ – Carlos Luna Feb 27 '15 at 12:47
  • $\begingroup$ OK. For most pure mathematicians logarithm means $log_e$. $\endgroup$ – tomi Feb 27 '15 at 13:01
  • $\begingroup$ No problem, I've specified somewhere in the text that I was working with decimal logarithms but I should definitely use a better notation (i.e. $\log_{10}$). Regarding your answer... How would you compute the antilogarithm of an arbitrary number using your proposed table? With a regular log table it can be done with an approximate reverse look-up but with a more sparse log table (like the one you propose, that mainly contains the logs of prime numbers) the interpolation errors should increase significantly, aren't they? $\endgroup$ – Carlos Luna Feb 27 '15 at 13:14
  • $\begingroup$ For five decimals you need a 250 values table ( based by theorem- it is the rest of lagrange interpolation). Supose you have a 10000 values table and you need log (5000.5).so log 5000=3.69897 and log 5001=3.69906 difference 9 by interpolation you deduce log 5000.5=69902 .... but correct rounded log in base 10 of 5000.5 is 69901! $\endgroup$ – florin Aug 1 '16 at 4:52
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I propose to use logs the base $10$.

Make two tables, one $x\mapsto y:=\log_{10} x$ with $x=[1.0 ..(0.5)..10]$, and one $y:=\log_{10}x\mapsto x$ with $y=[0..(0.05)..1]$.

Replace $0.5$ and $0.05$ by $0.2$ and $0.02$ if space is available.

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