Logarithms were introduced by John Napier in the early 17^{th} century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers and others to perform computations more easily, using slide rules and logarithm tables.
Logarithm tables are used to determine the logarithm of numbers. Let us now discuss how to find the logarithm (with base 10) of numbers.
Let \(x\) be a positive real number. Then, \(\text{log}\ x\) can be written as \(\text{log}\ x=\text{Characteristic of log}\ x +\text{Mantissa of log}\ x,\) where characteristic of \(\text{log}\ x\) is an integer that can be either positive or negative depending on whether \(x>1\) or \(0\lt x \lt 1\) and mantissa of \(\text{log}\ x\) has to be read from logarithm table. |
How to determine the characteristic of \(\text{log}\ x\)
How to determine the mantissa of \(\text{log}\ x\)
The mantissa has to be read from a standard logarithm table. Logarithm tables consist of rows headed by the values 10, 11, …, 99 and the columns headed by the values 0, 1, …, 9. Beyond these 10 columns, there is another set of 9 columns which give the mean difference. For determining the mantissa, the following has to be remembered:
Example 1: Find the value of \(\text{log}\ 405.6\). |
Solution: Let \(x=405.6\).
To find characteristic of \(\text{log}\ x\):
Since, \(x>1\) and the number of digits on the left of the decimal point is 3.
Therefore, characteristic \(=3-1=2\).
To find mantissa of \(\text{log}\ x\):
We ignore the decimal and read from the logarithm table a number 40 5 6. In the row headed by 40, choose the numbers in the column headed by 5 and the mean difference column headed by 6. The numbers are 6075 and 6 respectively.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean Difference | |||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |||||||||||
39 | 5911 | 5922 | 5933 | 5944 | 5955 | 5966 | 5977 | 5988 | 5999 | 6010 | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 10 |
40 | 6021 | 6031 | 6042 | 6053 | 6064 | 6075 | 6085 | 6096 | 6107 | 6117 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 9 | 10 |
41 | 6128 | 6138 | 6149 | 6160 | 6170 | 6180 | 6191 | 6201 | 6212 | 6222 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
The sum of these numbers \(=6075+6=6081\).
Therefore, mantissa \(=.6081\).
Hence, \(\text{log}\ 405.6=\text{Characteristic}+\text{Mantissa}=2+.6081=2.6081\).
Example 2: Find the value of \(\text{log}\ 0.00932\). |
Solution: Let \(x=0.00932\).
To find characteristic of \(\text{log}\ x\):
Since, \(0\lt x \lt 1\) and the number of digits on the left of the decimal point is 2.
Therefore, characteristic \(-(2+1)=-3\).
To find mantissa of \(\text{log}\ x\):
We ignore the decimal and read from the logarithm table a number 93 2 0. In the row headed by 93, choose the number in the column headed by 2 and take the mean difference as 0. The numbers are 9694 and 0.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean Difference | |||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |||||||||||
92 | 9638 | 9643 | 9647 | 9652 | 9657 | 9661 | 9666 | 9671 | 9675 | 9680 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
93 | 9685 | 9689 | 9694 | 9699 | 9703 | 9708 | 9713 | 9717 | 9722 | 9727 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
94 | 9731 | 9736 | 9741 | 9745 | 9750 | 9754 | 9759 | 9763 | 9768 | 9773 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
The sum of these numbers \(=9694+0=9694\).
Therefore, mantissa \(=.9694\).
Hence, \(\text{log}\ 0.00932=\text{Characteristic}+\text{Mantissa}=-3+.9694=-2.0306\).
Example 3: Find the value of \(\text{log}\ 13579\). |
Solution: Let \(x=13579\).
To find characteristic of \(\text{log}\ x\):
Since, \(x>1\) and the number of digits on the left of the decimal point is 5.
Therefore, characteristic \(=5-1=4\).
To find mantissa of \(\text{log}\ x\):
We ignore the decimal and read from the logarithm table a number 13 5 8 (which is obtained by rounding off the last digit of 13579). In the row headed by 13, choose the numbers in the column headed by 5 and the mean difference column headed by 8. The numbers are 6075 and 6 respectively. (Note: Rounding off should be done only after finding the characteristic.)
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean Difference | ||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||||||||
12 | 0792 | 0828 | 0864 | 0899 | 0934 | 3 | 7 | 11 | 14 | 18 | 21 | 25 | 28 | 32 | ||||||
0969 | 1004 | 1038 | 1072 | 1106 | 3 | 7 | 10 | 14 | 17 | 20 | 24 | 27 | 31 | |||||||
13 | 1139 | 1173 | 1206 | 1239 | 1271 | 3 | 6 | 10 | 13 | 16 | 19 | 23 | 26 | 29 | ||||||
1303 | 1335 | 1367 | 1399 | 1430 | 3 | 7 | 10 | 13 | 16 | 19 | 22 | 25 | 29 | |||||||
14 | 1461 | 1492 | 1523 | 1553 | 1584 | 3 | 6 | 9 | 12 | 15 | 19 | 22 | 25 | 28 | ||||||
1614 | 1644 | 1673 | 1703 | 1732 | 3 | 6 | 9 | 12 | 14 | 17 | 20 | 23 | 26 | |||||||
The sum of these numbers \(=1303+25=1328\).
Therefore, mantissa \(=.1328\).
Hence, \(\text{log}\ 13579=\text{Characteristic}+\text{Mantissa}=4+.1328=4.1328\).
Example 4: Find the value of \(\text{log}\ 0.035145\). |
Solution: Let \(x=0.035145\).
To find characteristic of \(\text{log}\ x\):
Since, \(0\lt x \lt 1\) and the number of zeroes appearing on the right side of the decimal point is 1.
Therefore, characteristic \(=-(1+1)=-2\).
To find mantissa of \(\text{log}\ x\):
We ignore the decimal and read from the logarithm table a number 35 1 5 (which is obtained by rounding off the last digit of 35145). In the row headed by 35, choose the numbers in the column headed by 1 and the mean difference column headed by 5. The numbers are 5453 and 6 respectively.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean Difference | |||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |||||||||||
34 | 5315 | 5328 | 5340 | 5353 | 5366 | 5378 | 5391 | 5403 | 5416 | 5428 | 1 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 |
35 | 5441 | 5453 | 5465 | 5478 | 5490 | 5502 | 5514 | 5527 | 5539 | 5551 | 1 | 2 | 4 | 5 | 6 | 7 | 9 | 10 | 11 |
36 | 5563 | 5575 | 5587 | 5599 | 5611 | 5623 | 5635 | 5647 | 5658 | 5670 | 1 | 2 | 4 | 5 | 6 | 7 | 8 | 9 | 11 |
The sum of these numbers \(5453+6=5459\).
Therefore, mantissa \(=.5459\).
Hence, \(\text{log}\ 0.035145=\text{Characteristic}+\text{Mantissa}=-2+.5459=-1.4541\).