INTRODUCTION TO TRIGONOMETRY.

LOGARITHMS.

1. The LOGARITHM of a given number is the exponent of the power to which it is necessary to raise a fixed numbe to produce the given number.

The fixed number is called THE BASE OF THE SYSTEM. Any positive number, except 1, may be taken as the base of a system. In the In the common system, to which alone reference is here made, the base is 10. Every number is, therefore, regarded as some power of 10, and the exponent of that power is the logarithm of the number.

2. If we denote any positive number by n, and the corresponding exponent of 10 by x, we shall have the exponential equation,

10x = n.

(1.)

In this equation, x is, by definition, the logarithm of n, which may be expressed thus,

3. If a number is an exact power of 10, its logarithm is a whole number. Thus, 100, being equal to 102, has for its logarithm 2. If a number is not an exact power of 10, its logarithm is composed of two parts, a whole number called the CHARACTERISTIC, and a decimal part called the MANTISSA. Thus, 225 being greater than 102 and less than 103, its logarithm is found to be 2.352183,

of which 2 is the characteristic and .852183 is the man tissa.

4. If, in the equation,

log (10)" = P,

(3.)

we make p successively equal to 0, 1, 2, 3, &c., and also

equal to lowing

- 3, &c., we may form the fol

If a number lies between 1 and 10, its logarithm lies between 0 and 1, that is, it is equal to 0 plus a decimal; if a number lies between 10 and 100, its logarithm is equal to 1 plus a decimal; if between 100 and 1000, its logarithm is equal to 2 plus a decimal; and so on; hence, we have the following

RULE. The characteristic of the logarithm of an entire number is positive, and numerically 1 less than the number of places of figures in the given number.

If a decimal fraction lies between .1 and 1, its logarithm lies between 1 and 0, that is, it is equal to -1 plus a decimal; if a number lies between .01 and .1, its logarithm is equal to 2 plus a decimal; if between .001 and .01, its logarithm is equal to 3 plus a decimal; and so on: hence, the following

RULE-The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of O's that immediately follow the decimal point.

The characteristic alone is negative, the mantissa being always positive. This fact is indicated by writing the negative sign over the characteristic: thus, 2.371465, is equivalent to 2.371465.

NOTE. It is to be observed, that the characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus, the characteristic of the logarithm of 725.4275 is the same as the characteristic of the logarithm of 725.

GENERAL PRINCIPLES.

5. Let m and n denote any two numbers, and x and y their logarithms. We shall have, from the definition of a logarithm, the following equations,

Multiplying (4) and (5), member by member, we have

That is, the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

6. Dividing (4) by (5), member by member, we have

That is, the logarithm of a quotient is equal to the loga. rithm of the dividend diminished by that of the divisor.

7. Raising both members of (4) to the power denoted by p, we have,

That is, the logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

8. Extracting the root, indicated by r, of both members of (4), we have

That is, the logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root.

The preceding principles enable us to abbreviate the operations of multiplication and division, by converting them into the simpler ones of addition and subtraction.

TABLE OF LOGARITHMS.

9. A TABLE OF LOGARITHMS is a table containing a set of numbers and their logarithms, so arranged that,, having given any one of the numbers, we can find its logarithm; or, having the logarithm, we can find the corresponding number.

In the table appended, the complete logarithm is given for all numbers from 1 up to 100. For other numbers,

the mantissas alone are given; the characteristic may be found by one of the rules of Art. 4.

Before explaining the use of the table, it is to be shown that the mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact power of 10.

Let ʼn represent any number whatever, and 10" any power of 10, p being any whole number, either positive or negative. Then, in accordance with the principles of Arts. 5 and 3, we shall have

log (n × 10o) = log n + log 10" = p + log n;

but p is, by hypothesis, a whole number: hence, the decimal part of the log (n x 10") is the same as that of log n; which was to be proved.

Hence, in finding the mantissa of the logarithm of a number, the position of the decimal point may be changed at pleasure. Thus, the mantissa of the logarithm of 456357, is the same as that of the number 4563.57; and the mantissa of the logarithm of 759 is the same as that of 7590.

MANNER OF USING THE TABLE.

1o. To find the logarithm of a number less than 100.

10. Look on the first page, in the column headed "N,” for the given number; the number opposite is the logarithm required. Thus,

log 67 1.826075.