Uses for logarithmic values
Logarithmic values are extremely useful in finding products, quotients and in extracting roots of a higher order than 3.
For example, find the numerical values of
Logarithmic values can also be used in changing the base of a logarithm.
Logarithmic tables
Logarithmic tables are of the form given below.

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
Mean Differences 
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 
26 
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 
16 
19 
23 
26 
29 
14 
1461 
1492 
1523 
1553 
1584 
1614 
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 
16 
2041 
2068 
2095 
2122 
2148 
2175 
2201 
2227 
2253 
2279 
3 
5 
8 
11 
13 
16 
18 
21 
24 
17 
2304 
2330 
2355 
2380 
2405 
2430 
2455 
2480 
2504 
2529 
2 
5 
7 
10 
12 
15 
17 
20 
22 
18 
2553 
2577 
2601 
2625 
2648 
2672 
2695 
2718 
2742 
2765 
2 
5 
7 
9 
12 
14 
16 
19 
21 
19 
2788 
2810 
2833 
2856 
2878 
2900 
2923 
2945 
2967 
2989 
2 
4 
7 
9 
11 
13 
16 
18 
20 
20 
8010 
3032 
3054 
3075 
3096 
3118 
3139 
3160 
3181 
3201 
2 
4 
6 
8 
11 
13 
15 
17 
19 
21 
3222 
3243 
3263 
3284 
3304 
3324 
3345 
3365 
3385 
3404 
2 
4 
6 
8 
10 
12 
14 
16 
18 
22 
3424 
3444 
3464 
3483 
3502 
3522 
3541 
3560 
3579 
3598 
2 
4 
6 
8 
10 
12 
14 
15 
17 
23 
3617 
3636 
3655 
3674 
3692 
3711 
3729 
3747 
3766 
3784 
2 
4 
6 
7 
9 
11 
13 
15 
17 
24 
3802 
3820 
3838 
3856 
3874 
3892 
3909 
3927 
3945 
3962 
2 
4 
5 
7 
9 
11 
12 
14 
16 
25 
3979 
3997 
4014 
4031 
4048 
4065 
4082 
4099 
4166 
4133 
2 
3 
5 
7 
9 
10 
12 
14 
15 
26 
4150 
4166 
4183 
4200 
4216 
4232 
4249 
4265 
4281 
4298 
2 
3 
5 
7 
8 
10 
11 
13 
15 
27 
4314 
4330 
4346 
4362 
4378 
4393 
4409 
4425 
4440 
4456 
2 
3 
5 
6 
8 
9 
11 
13 
14 
28 
4472 
4481 
4502 
4518 
4533 
4548 
4564 
4879 
4594 
4609 
2 
3 
5 
6 
8 
9 
11 
12 
14 
In the above table, the first column denotes the first two significant figures of the given number whose log is to be found. The next set of columns with 0,1 .... 9 at the head of the columns denotes the third significant figures of the numbers.
The numbers given under the heads of the first eleven columns are the mantissa of the logarithms with the decimal point omitted.
The numbers given under the heading “Mean Differences” are the approximate increments in the mantissa on account of the fourth significant figure in the given number.
Example 1
Find the logarithm of 24.
Solution:
The number consists of two digits before the decimal point since 24 = 24.0000.
Characteristic of log 24 = 21 = 1.
As the third and fourth significant figures of the given number are zeros, the mantissa of this number is the number in the row containing 24 and in the column headed by 0. So, from the table
log 24 = 1.3802
Example 2
Find the logarithm of 193.
Solution:
The number consists of three digits.
Characteristic of log 193 = 31 = 2.
The third significant figure of the number is 3 and the fourth is 0, the mantissa of 193 is that given in the row containing 19 and in the column headed by 3. From the table, this is 2856.
log 193 = 2.2856
Example 3
Find the logarithm of 2147
The number has four digits.
The characteristic of log 2147 = 41 = 3
The third significant figure is 4
The fourth significant figure is 7
The mantissa of the number is that given in the row containing 21 and in the column headed by 4, it is increased by the number given under the head 7 of the mean difference, with 7 being the fourth significant figure.
So the Mantissa is 3304 + 14 = 3318
log 2147 = 3.3318
Example 4
Find the logarithm of 2.356
There is only one digit before the decimal point.
The characteristic of log 2.356 = 11 = 0.
The numbers 2.356 and 2356 have the same significant figures and hence their mantissae are the same.
So the mantissa is found in the row containing 23 under the column headed by 5 for the third significant figure and is increased by the number given under the head 6 of the mean difference, 6 being the fourth significant figure.
So the mantissa is 3711 + 11 = 3722
or log 2.356 = 0.3722
Example 5
Example 6
Find the logarithm of 240562
In obtaining a logarithm, we need four significant figures. If a number has more than four significant figures, we round off the fourth digit to the nearest integer. So when we need to find the mantissa, 240562 becomes 2406.
The number has six digits.
The characteristic of log 240562 = 61 = 5.
The mantissa is obtained in the row containing 24 under the column headed by 0 and is increased by the number given under the head of 6 of the mean difference.
The mantissa is 3802 + 11 = 3813.
log 240562 = 5.3813
Try these questions
Find the logarithms of the following
 6.183
 786.24
 21.978
 0.6432
 0.0000787
Answers

Let x = 6.183
log x = log 6.183
= 0.7912
 Let x = 786.24
log x = log 786.24 = log 786.2
= 2.8955
 Let x = 21.978
log x = log 21.978 = log 21.98
= 1.3404 + 16 (mean difference)
log x = 1.3420