# How to use "modern" log tables

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

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

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).

An extract from a 1966 table of logarithms
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

### Some examples of the use of the table

#### A: Multiply 15.27 by 48.54

1. Look up the logarithms as before, the log of 15.27 is 1.1838 and the log of 48.54 is 1.6861
```     1.1838
+ 1.6861
---------
2.8699
```
3. At this point we have the logarithm of the answer - so what now?
4. 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.
An extract from a 1966 table of anti-logarithms
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
1. Mentally remove and store the characteristic (2)
2. Run index finger down the left-hand column until it finds .86
3. Move index finger along the row until it is on column 9 (it should now be over 7396)
4. Move other finger to difference column 9 and read the difference (15)
```    7396
+  15
------
7411
```
6. The characteristic was 2, meaning the answer is between 100 and 1000 so ...
7. 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.

#### B: Divide 54.19 by 10.27

To divide we have to subtract the logarithms so:-

1. Look up the logarithms
2. Subtract them
```     1.7339
- 1.0115
--------
0.7224
```
3. Then look up the antilog of .7224, (which is 5277)
4. Remembering that the characteristic is 0, insert the decimal point in the correct place 5.277

This time the result agrees with a calculator!

#### C: Find the square root of 52.73

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

1. Look up the logarithm of 52.73 (1.7220)
2. Divide by 2. (1.7220  ÷ 2 = 0.8610)
3. Look up the antilog of .8610 (7261)
4. 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, 