Logarithms are particularly useful in the practical calculations required in Trigonometry, Plane and Spherical, in Land-surveying, in Compound Interest and Annuities, in Astronomy, and in Navigation.

Other definitions and explanations will be given in the next chapter.

CHAPTER II.

ON THE USE OF LOGARITHMS.

PROBLEM I.

To Find, in Table I,* the Logarithm of any Integral
Number from 1 to 250.

RULE. Look for the number in the column headed N ; and adjacent to the number will be found its logarithm with the proper index.

NOTE 1. In this and subsequent problems, it will be understood that the logarithms in the Tables are not the exact logarithms of the numbers given, but merely true to the last decimal figure; for all logarithms, except those of the base and its powers, are interminate decimals.

EXERCISES. What are the logarithms of 5, of 56, and of 231; of 7 and of 70; of 1, of 10, and of 100; of 2, of 20, and of 200; of 23 and of 230?

NOTE 2. In resolving the preceding exercises, the scholar must have observed that the logarithms of 7 and of 70 are alike in their decimal part, and differ only in the integer. He may observe the same thing of 1, 10, and 100; of 2, 20, and 200; and of 23 and 230. This property is peculiar to Briggs's system, and is the cause of that system's being preferred to every other for common purposes. The principle on which the property depends is very simple. Since 20, for instance, is ten times 2, the logarithm of 20 will be the logarithm of 10 added to the logarithm of 2, from what

*Table I may be used for the same purpose; but, in general, Table I will be more convenient.

has been said in the last chapter. But the logarithm of 10 is 1. Therefore, the logarithm of 20 will be a unit more than that of 2. For the same reason, the logarithm of 200 will be a unit more than that of 20; that of 2000 will be a unit higher still; and so on. For the same reason, also, the logarithm of 230 is a unit more than that of 23; and the logarithm of 74 is a unit more than that of 7.4.

Hence, since the logarithm of 1 is 0, and that of 10 is 1, it follows that the logarithm of 100 is 2, that of 1000 is 3, that of 10,000 is 4, &c. &c. Hence, also, the logarithms of all numbers between 1 and 10 will be between 0 and 1, or will be 0 with a decimal fraction; those of numbers between 10 and 100 will be 1 with a decimal; between 100 and 1000, 2 with a decimal; and so on.

DEFINITION. The integral part of the logarithm is called its Index.

NOTE 3. From what has just been said, it follows that the index of all numbers between 1 and 10 will be 0; that of numbers between 10 and 100, 1; between 100 and 1000, 2, &c. Or the index will always be a unit less than the number of integral figures in the natural number. Thus the index of 4 will be 0; that of 64 will be 1; that of 365, 2; that of 4821, 3; that of 56908, 4; and so on. Hence the Rule for the following Problem.

PROBLEM II.

To find the Index of the Logarithm of any given Number, partly integral, partly decimal.

RULE. Count the number of integral figures. That number, diminished by 1, will be the index.

Thus the logarithmic index of the number 7, of 7·5, of 6.28, or of 3.084, is 0; the index of 63, of 26.7, or of 94-108, is 1; that of 865-24, is 2; that of 7654, or of 5836 1, is 3; and so on.

EXERCISE. What are the logarithmic indices of the natural numbers 5, 76, 3920, 69-4, 706, 521-85, 1.6, 3.007, and 581,296?

NOTE.

When the given number is a decimal fraction without integers, the index is negative and is written thus, 2. We observe in what place of decimals, in the given number, the first significant figure occurs, and the number of that place, with the negative sign written over it, is the index. Thus the index of 582 is 1; that of 058 is 2; that of 006 is 3; and so on. But, since the negative indices are troublesome to beginners, they will be dispensed with altogether in this volume, and rules given for working every question without them.

PROBLEM IV.

To find, in Table II, the Logarithm of any number consisting of not more than three Figures.

NOTE 1. When we say, in this place, "three figures," we mean independently of ciphers either to the right or to the left. Thus we should include 6340, 73000, and 00265 under the head of this problem.

RULE. If the number consists of three figures, look for it in the left-hand column of the table; and opposite it, in the next column, will be found its logarithm, but without the index. The index must be found by Problem II, and inserted.

Thus the logarithm of 576 is 2-760422; that of 39-4 is 1.595496; that of 0253 is 2.403121.

If the number consists of less than three figures,* make up three by placing ciphers, if not already present (or by supposing them placed), on the right of the number, such ciphers being always regarded as decimals. Then proceed as before.

Thus the logarithm of 75 is the same as that of 75.0 the logarithm of 8 is the same as of 8.00; and that of '035 is the same as of 0350.

NOTE 2. In both cases a decimal point occurring in the given number is taken no notice of till we come to the insertion of the index.

In this case the logarithm may be found also from Table 1, by changing the index, if necessary.

NOTE 3. When a blank occurs in the left side of the column of logarithms, its place must be filled up by taking in the two figures next above it in the same column.

EXERCISE 1. What are the logarithms of 563, of 708, and of 550?

2. What of 365, of 206, and of 900? (See Note 3). 3. Find the logarithms of the numbers 24.6, and 3·88. (See Note 2).

4. Find the logarithms of 8,540 and of 73,000. (See Note 1).

5. What are the logarithms of 27, of 3.9, and of 8·0?

PROBLEM V.

To find, from Table II, the Logarithm of any Number represented by four Figures, integral or decimal.

RULE. Look for the three first figures of the number in the first column of the table. Having found these, we have the line in which the required logarithm is to be found. Then look for the fourth figure of the given number in the line of digits at the top or at the bottom of the same page. This will show the column in which the required logarithm is contained. Let the eye then move along the line previously found till it come to the proper column, it will then rest on the logarithm sought, or rather on the four last figures of that logarithm. The two other figures are to be taken from the left-hand side of the lefthand column of logarithms, taking the two at the beginning of the same line, unless their place is blank; but, if blank, then the two next above the blank or the two next below it. The two next above are always to be taken unless the first of the four figures previously found has a hyphen* over it, thus 2; in which case we take the two next below. Having written these two figures before the four previously found, prefix the decimal point, and the index, found as before.†

* The mark is the same as that for the negative sign over an index; but, from its position, it can never be mistaken for it.

†The teacher will probably find it necessary to explain this rule to the learner, and to see him practise it. The rule seems complicated, but becomes easy enough after the first trial.

Thus the logarithm of 3470 is 3.540329; the logarithm of 3478 is 3.541330; that of 3·492 is 0·543074; and that of 0.3468 is 1.540079.

NOTE. Observe Notes 1 and 2 of the last problem.

EXERCISE 1. What are the logarithms of the numbers 4580, 4586, 87-40, 87·42, 3.357, and 933.4?

2. What of 49,210, of 7;836,000, of 380,800, and of 72;000,000?

3. Find the logarithms of 25,860, of 2,586, of 258-6, of 25.86, and of 2.586.

PROBLEM VI.

To find the Logarithm of any Number represented by more than four Figures.

RULE. Regard all the figures after the first four as ciphers, and find the logarithm by Problem v.

If the first of the neglected figures is greater than 5, or if it is 5 followed by other significant figures, the preceding figure should be increased by a unit on dropping those that follow.*

Thus, instead of 54682, we look for 54680; and, instead of 93.6481, we look for 93.65.

EXERCISE. Find the logarithms of the numbers 93,514; 48,627; 78,362; 9,244-8; 3.2764; 832,962; 12,954-8; 300-424; and 168;544,919.

Answers: 4.970858; 4.686904; 1.894094; 3.965907; 0.515344; 5.920645; 4·112270; 2·477700; and 8.226600.

PROBLEM IX.

To find the Natural Number answering to any given
Logarithm.

RULE. Observe the two figures first after the index, and look for them in the left margin of the first column of

This rule is sufficiently accurate for all questions in an elementary course. For those requiring more accurate calculation, the student is referred to Rule II of this Problem in the Author's larger work on the subject.