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 = 2-1 = 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 = 3-1 = 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 = 4-1 = 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 = 1-1 = 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 = 6-1 = 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

1. 6.183
2. 786.24
3. 21.978
4. 0.6432
5. 0.0000787

Answers

1. Let x = 6.183
   log x = log 6.183
             = 0.7912


2. Let x = 786.24
   log x = log 786.24 = log 786.2
           = 2.8955


3. Let x = 21.978
   log x = log 21.978 = log 21.98
             = 1.3404 + 16        (mean difference)
   log x = 1.3420