INTRODUCTION TO TRIGONOMETRYYORK

PUBLIC LIBRARY

LOGARITHMS.

ASTOR, LENOX AND TILDEN FOUNDATIONS. 1899

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 n, and the corresponding exponent of 10, by x, we shall have the exponential equation,

In this equation, x 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.

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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

-

If a number lies between 1 and 10, its logarithm lies between 0 and 1, that is, it is equal to 0 plus a deci mal; 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 and so on: hence, the following

- 3, plus a decimal ;

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 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 a and y their logarithms. We shall have, from the defini

tion of a logarithm, the following equations,

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

10+ y = mn;

whence, by the definition,

x + y = log (mn).

(6.)

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

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.