### How to use Logarithm Table

Logarithms were introduced by John Napier in the early 17th 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 $\text{log}\ x$

• If $x>1$, then count the digits on the left of the decimal point. If the number of digits is $n$, then the characteristic is $(n-1)$.
• If $0\lt x \lt 1$, then count the number of zeros appearing on the right side of the decimal point. If the number of zeros is $n$, then the characteristic is $-(n+1)$.

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:

• While determining the mantissa, the decimal point of the number has to be ignored.
• Mantissa is calculated for a four-digit number.
• If the number of digits is more than 4, then it is rounded off to get a four-digit number.
• Two numbers have to be chosen from a particular row appearing in two particular columns. The first two digits represent the row number, the third digit represents the column number and the fourth digit represents the mean difference column number. If the fourth digit is 0, we take the mean difference as 0.
• Mantissa is the sum of two numbers with a decimal placed before the first digit.

 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, $\text{log}\ 0.035145=\text{Characteristic}+\text{Mantissa}=-2+.5459=-1.4541$.