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B How reliable are logarithm tables?

  1. Feb 14, 2017 #1
    Today I came across a high school math book which has a particular problem in the logarithms chapter. It has
    $$ \log_{10}{0.2913} = -1.4643 $$
    Trying to verify it with a calculator, I get -0.53566. There's a log table attached at the end which agrees with the calculation made in the book. To make sure there wasn't a typo, I looked up online for the common logarithm table and found tables that agree with it. Trying to verify the book's calculation, I got (with a calculator)
    $$ 10^{-1.4643} = 0.034332 $$
    Now am I missing something or is it something wrong with the logarithm tables I have? Admittedly, it has been a very long time since I last calculated logarithms using a table.
    Last edited: Feb 14, 2017
  2. jcsd
  3. Feb 14, 2017 #2


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    That is a very odd error, as it is off by nearly (but not exactly) 1. Are the logarithm values next to it wrong in a similar way?
    Does your logarithm table really have exactly this entry?
  4. Feb 14, 2017 #3
    It is off for the next value in the same problem too. It calculates $$ \log_{10}{0.004236} $$ as -3.6269 while with a calculator I get -2.373044.
    Even weirder is that it proceeds to add the two logarithms (which it calculated as -1.4643 and -3.6269) to get -3.0912. Then it proceeds to take the anti-log of -3.0912 and gets 0.001234 (while with a calculator I get $$ 10^{-3.0912} = 0.0008106 $$). At this point I stopped taking the book seriously but thought I'd make sure whether it's a problem with me or the book before I explain it to my friend. o0)
  5. Feb 14, 2017 #4


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    It's written in what we called 'bar' notation in school .

    Roughly : number of decade shifts + basic log of number in range 1 to 10

    0.4643 is the log of 2.913

    log of 0.2913 = -1 + 0.4643 = - 0.5357

    You can see why this works :

    log 0.2913 = log 2.913 - log 10
  6. Feb 14, 2017 #5
    Oh. So the bar isn't actually a negative? Why would they decide to write it this way? o_O
    Last edited: Feb 14, 2017
  7. Feb 14, 2017 #6
    Are there cases where the bar notation is useful other than confusing poor unsuspecting readers?
  8. Feb 14, 2017 #7


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    It greatly reduces the number of log values that are needed in tables . All you need are logs for numbers in range 1 to 10 .

    There are also some small advantages in making calculations more systematic and in reducing chance of order of magnitude errors in final answers .
  9. Feb 14, 2017 #8


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    Conventionally written down like this :
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