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
101 = 10
102 = 100
103 = 1000
104 = 10000
105 = 100000
106 = 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
104 × 102 = 106
(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 Log10). 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 (1-1/107).
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.something 150 lies between 100 and 1000 so its logarithm must be between 2 and 3, ie it is 2.something 1500 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 MANTISSA.
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
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 |
1.1838 + 1.6861 --------- 2.8699
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 |
7396 + 15 ------ 7411
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:-
1.7339 - 1.0115 -------- 0.7224
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)
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,