CA

TRIGONOMETRY.

BOOK I.

THE NATURE AND PROPERTIES OF LOGARITHMS.

ARTICLE 1. Logarithms are numbers designed to diminish the labor of Multiplication and Division, by substituting in their stead Addition and Subtraction. All numbers are regarded as powers of some one number, which is called the base of the system; and the exponent of that power of the base which is equal to a given number, is called the logarithm of that number.

The base of the common system of logarithms (called, from their inventor, Briggs' logarithms) is the number 10. Hence all numbers are to be regarded as powers of 10. Thus, since 10°=1, O is the logarithm of 1 in Briggs' system

whence it appears that, in Briggs' system, the logarithm of every number between 1 and 10 is some number between 0 and 1, i. e., is a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2, i. e., is 1 plus a fraction. The logarithm of The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e., is 2 plus

a fraction, and so on.

(2.) The preceding principles may be extended to fractions by means of negative exponents. Thus, since

10=0.1, -1 is the logarithm of 0.1

10-2=0.01, -2 66 10-0.001, −3

in Briggs' system;

66

0.01

66

66

66

0.001

Hence it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1, or may be represented by 1 plus a fraction; the logarithm of every number between 0.1 and .01 is some number between −1 and -2, or may be represented by -2 plus a fraction; the logarithm of every number between .01 and .001 is some number between 2 and 3, or is equal to -3 plus a fraction, and

so on.

The logarithms of most numbers, therefore, consist of an integer and a fraction. The integral part is called the characteristic, and may be known from the following

RULE.

The characteristic of the logarithm of any number greater than unity, is one less than the number of integral figures in the given number.

Thus the logarithm of 297 is 2 plus a fraction; that is, the characteristic of the logarithm of 297 is 2, which is one less than the number of integral figures. The characteristic of the logarithm of 5673.29 is 3; that of 73254.1 is 4, &c.

The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which its first significant figure is removed from the place of units.

Thus the logarithm of .0046 is -3 plus a fraction; that is, the characteristic of the logarithm is -3, the first significant figure, 4, being removed three places from units.

(3.) Since powers of the same quantity are multiplied by adding their exponents (Alg., Art. 50),

The logarithm of the product of two or more factors is equal to the sum of the logarithms of those factors.

Hence we see that if it is required to multiply two or more numbers by each other, we have only to add their logarithms: the sum will be the logarithm of their product. We then look in the table for the number answering to that logarithm, in order to obtain the required product.

Also, since powers of the same quantity are divided by subtracting their exponents (Alg., Art. 66),

The logarithm of the quotient of one number divided by an