INTRODUCTION TO TRIGONOMETRY.

LOGARITHMS.

1. THE LOGARITHM of a number is the exponent of the power to which it is necessary to raise a fixed number, 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 common system, the base is 10.

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

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

3. From the definition of a logarithm, it follows that, the logarithm of any power of 10 is equal to the exponent of that power: hence the formula,

If a number is an exact power of 10, its logarithm is

a whole number.

If a number is not an exact power of 10, its logarithm will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally.

The entire part of a logarithm is called the characteristic, the decimal part, is called the mantissa.

4. If, in Equation (3), we make p successively equal to 0, 1, 2, 3, &c., and also equal to 0, 1, -2, -3, &c., we may form the following

log 1 = 0

log 10 = 1

TABLE.

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log .01 =

= 3

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,

-1

If a decimal fraction lies between 1 and 1, its loga rithm lies between 1 and 0, that is, it is equal to 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 0'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 equiv alent to 2 + .371465.

It is to be observed, that the characteristic cf the logarithm of a mixed number is the same as that of its entire part. Thus, the mixed number 74.103, lies between 10 and 100; hence, its logarithm lies between 1 and 2, as does the logarithm of 74.

5. Let m and n denote any two numbers, and

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,

10=P = m2;

whence, by the definition,

xp = log m3.

(8.)

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 oper ations 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 10,000. 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 n 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 x 10") = log n + log 10o = p + log n ;

but p is, by hypothesis, a whole number: hence, the deci mal 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, we may regard the number as a decimal, and move the decimal point to the right or left, 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 2.00357, is the same as that of 2003.57.