# B How reliable are logarithm tables?

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1. Feb 14, 2017

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. Feb 14, 2017

### Staff: Mentor

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?

3. Feb 14, 2017

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.

4. Feb 14, 2017

### Nidum

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

5. Feb 14, 2017

Oh. So the bar isn't actually a negative? Why would they decide to write it this way?

Last edited: Feb 14, 2017
6. Feb 14, 2017

Are there cases where the bar notation is useful other than confusing poor unsuspecting readers?

7. Feb 14, 2017

### Nidum

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 .

8. Feb 14, 2017

### Nidum

Conventionally written down like this :