This box describes the so-called four figure tables, which are the easiest to use and suitable for most calculations. The more accurate six and seven figure tables are more difficult to describe and were never in common usage.

First of all, thinking about powers of 10, we can write

10^{1} = 10

10^{2} = 100

10^{3} = 1000

10^{4} = 10000

10^{5} = 100000

10^{6} = 1000000

and so on...

Now, if we multiply ten thousand by one hundred we get one million, or writing this down:-

10,000 × 100 = 1,000,000

0r

10^{4} × 10^{2} = 10^{6}

(Notice 4 + 2 = 6)

In other words, we can **multiply** two numbers
by **adding** their powers. These powers are the
**logarithms** of the numbers. For instance, the
logarithm of 100 is 2.

This is the modern way of describing logarithms, they are
**powers** of a particular **base**. In
the case we are looking at, the base is 10. (Sometimes written
Log_{10}). Note that this is different to the original
Napier Logarithms, which were derived by repeated multiplication
by a number close to, but smaller than, one
(1-1/10^{7}).

What about the logarithm of 101 or 15.27?

This is where the tables come in, they contain all the intermediate values to fill in the gaps between 10 and 100 and 100 and 1000 and so on.

To make the tables a manageable size, they do not contain the logarithms of all possible numbers, instead, the logarithm is split into two parts, separating the size of the number from its value. For example:-

Considering the numbers 15, 150 and 1500 15 lies between 10 and 100 so its logarithm must be between 1 and 2, ie it is 1.something150 lies between 100 and 1000 so its logarithm must be between 2 and 3, ie it is 2.something1500 lies between 1000 and 10000 so its logarithm must be between 3 and 4, ie it is 3.something

In each case the
"** .something**" has the same
value and this is what is printed in the tables. It is known as
the

YOU have to supply the number (1,2,3 ...) that goes before the
decimal point. It is known as the *CHARACTERISTIC*.

In the tables, the logarithm of 15 is printed as 1761, so for our examples,

The logarithm of 15 is 1.1761 The logarithm of 150 is 2.1761 The logarithm of 1500 is 3.1761 and so on...

The table below is an extract from a book of 4 figure log tables, published in 1966. Note that all the logarithms are decimal fractions, but the decimal point is not printed. This is a long standing tradition. Differences refer to the least significant figures (rightmost) and so should have zeros in front of them, again these are understood to be there, they are not printed. This does, in fact, make the tables more readable but takes some getting used to.

To look up a logarithm of say, 15**.**27, you
would

- First work out the characteristic, in this case it is 1
- Run your index finger down the left-hand column until it reaches 15
- Now move it right until it is on column 2 (it should be over 1818)
- Using another finger, find the difference on column 7 of the differences (20)
- Add the difference.

So the logarithm of 15**.**27 is
1**.**1818 + 0**.**0020 =
1**.**1838

All the "fingering" of the table is the reason that surviving log tables often have a grubby, brown and rather greasy look (mine do, anyway).

L O G A R I T H M S | ------ Mean Differences ------ | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

10 | 0000 | 0043 | 0086 | 0128 | 0170 | 0212 | 0253 | 0294 | 0334 | 0374 | 4 | 8 | 12 | 17 | 21 | 25 | 29 | 33 | 37 |

11 | 0414 | 0453 | 0492 | 0531 | 0569 | 0607 | 0645 | 0682 | 0719 | 0755 | 4 | 8 | 11 | 15 | 19 | 23 | 28 | 30 | 34 |

12 | 0792 | 0828 | 0864 | 0899 | 0934 | 0969 | 1004 | 1038 | 1072 | 1106 | 3 | 7 | 10 | 14 | 17 | 21 | 24 | 28 | 31 |

13 | 1139 | 1173 | 1206 | 1239 | 1271 | 1303 | 1335 | 1367 | 1399 | 1430 | 3 | 6 | 10 | 13 | 18 | 19 | 23 | 26 | 29 |

14 | 1461 | 1492 | 1523 | 1553 | 1584 | 1814 | 1644 | 1673 | 1703 | 1732 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |

15 | 1761 | 1790 | 1818 | 1847 | 1875 | 1903 | 1931 | 1959 | 1987 | 2014 | 3 | 6 | 8 | 11 | 14 | 17 | 20 | 22 | 25 |

- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |

- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |

46 | 6628 | 6637 | 6646 | 6656 | 6665 | 6675 | 6684 | 6693 | 8702 | 6712 | 1 | 2 | 3 | 4 | 5 | 8 | 7 | 7 | 8 |

47 | 6721 | 6730 | 6739 | 6749 | 6758 | 6767 | 6776 | 6785 | 6794 | 6803 | 1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 | 8 |

48 | 6812 | 6821 | 6830 | 6839 | 6848 | 6857 | 6866 | 6875 | 6884 | 6893 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 8 |

49 | 6902 | 6911 | 6920 | 6928 | 6937 | 6946 | 6955 | 6964 | 6972 | 6981 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 8 |

50 | 6990 | 6998 | 7007 | 7016 | 7024 | 7033 | 7042 | 7050 | 7059 | 7067 | 1 | 2 | 3 | 3 | 4 | 5 | 6 | 7 | 8 |

- Look up the logarithms as before, the log of
15
**.**27 is 1**.**1838 and the log of 48**.**54 is 1**.**6861 - Add the logarithms
1.1838 + 1.6861 --------- 2.8699

- At this point we have the logarithm of the answer - so what now?
- It is possible to use the log tables backwards, but most people would have turned to the next page for the table of antilogarithms - printed below.

A N T I L O G A R I T H M S | ----- Mean Differences ----- | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

.70 |
5012 | 5023 | 5035 | 5047 | 5058 | 5070 | 5082 | 5093 | 5105 | 5117 | 1 | 2 | 4 | 5 | 6 | 7 | 8 | 9 | 11 |

.71 |
5129 | 5140 | 5152 | 5164 | 5178 | 5188 | 5200 | 5212 | 5224 | 5236 | 1 | 2 | 4 | 5 | 6 | 7 | 8 | 10 | 11 |

.72 |
5248 | 5260 | 5272 | 5284 | 5297 | 5309 | 5321 | 5333 | 5348 | 5358 | 1 | 2 | 4 | 5 | 8 | 7 | 9 | 10 | 11 |

.73 |
5370 | 5383 | 5395 | 5408 | 5420 | 5433 | 5445 | 5458 | 5470 | 5483 | 1 | 3 | 4 | 5 | 8 | 8 | 9 | 10 | 11 |

.74 |
5495 | 5508 | 5521 | 5534 | 5546 | 5559 | 5572 | 5585 | 5598 | 5610 | 1 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 12 |

.75 |
5623 | 5636 | 5649 | 5662 | 5675 | 5689 | 5702 | 5715 | 5728 | 5741 | 1 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | 12 |

- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |

- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |

.86 |
7244 | 7261 | 7278 | 7295 | 7311 | 7328 | 7345 | 7382 | 7379 | 7396 | 2 | 3 | 5 | 7 | 8 | 10 | 12 | 13 | 15 |

.87 |
7413 | 7430 | 7447 | 7464 | 7482 | 7499 | 7516 | 7534 | 7551 | 7568 | 2 | 3 | 5 | 7 | 9 | 10 | 12 | 14 | 16 |

.88 |
7586 | 7603 | 7621 | 7638 | 7656 | 7674 | 7691 | 7709 | 7727 | 7745 | 2 | 4 | 5 | 7 | 9 | 11 | 12 | 14 | 16 |

.89 |
7782 | 7780 | 7798 | 7816 | 7814 | 7852 | 7870 | 7889 | 7907 | 7925 | 2 | 4 | 5 | 7 | 9 | 11 | 13 | 14 | 16 |

.90 |
7943 | 7962 | 7980 | 7998 | 8017 | 8035 | 8054 | 8072 | 8091 | 8110 | 2 | 4 | 6 | 7 | 9 | 11 | 13 | 13 | 17 |

- Mentally remove and store the characteristic (2)
- Run index finger down the left-hand column until it finds
**.86** - Move index finger along the row until it is on column 9 (it should now be over 7396)
- Move other finger to difference column 9 and read the difference (15)
- Add the difference.
7396 + 15 ------ 7411

- The characteristic was 2, meaning the answer is between 100 and 1000 so ...
- Insert the decimal point in the correct place
**741.1**

Try this on a calculator - the answer is different,
741**.**2. This is often the case, because of
rounding errors. Log tables were faster than hand methods but
four figure tables could only be relied on for three figure
accuracy.

To divide we have to subtract the logarithms so:-

- Look up the logarithms
- Subtract them
1.7339 - 1.0115 -------- 0.7224

- Then look up the antilog of .7224, (which is 5277)
- Remembering that the characteristic is 0, insert the decimal
point in the correct place
**5.277**

This time the result agrees with a calculator!

Square roots are formed by dividing the logarithm by 2 (for cube roots, divide by 3 ... etc)

- Look up the logarithm of 52.73 (1.7220)
- Divide by 2. (1.7220 ÷ 2 = 0.8610)
- Look up the antilog of .8610 (7261)
- Insert the decimal point in the correct place
**7.261**

This time the calculator gives 7.2615, so the result should be 7.262

Finally,

I hope that you have found this page useful,

good luck,

29-Jan-2002

Revised 18-Oct-2009

Text Copyright © 2002 A. Audsley, All Rights Reserved

Text Copyright © 2002 A. Audsley, All Rights Reserved