CHAPTER XI.

LOGARITHMS.

Definitions.

185. 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.

186. If we denote any positive number, except 1, by a, any positive number whatever by n, and the exponent of the power to which it is necessary to raise a, in order to produce n, by x, we shall have the exponential equation,

In this equation, a is the base, n any positive number, and x is the logarithm of n. It is plain, that a cannot be negative, neither can it be equal to 1, because every power of 1 is equal to 1.

If, whilst a remains fixed in value, we suppose n to assume in succession every value from 0 to, the corresponding values of x, taken together, will constitute what is called a system of logarithms. Since there are an infinite number of different values that

may be attributed to a, it follows, that there are an infinite number of systems of logarithms. Of these, two only are in general use, viz.: the system whose base is 10, called the common system; and the system whose base is 2.718281828..., called the Napierian system.

In what follows, we shall designate common logarithms by the symbol log, Napierian logarithms by the symbol 1, and logarithms taken in any system whatever, by the symbol Log.

187. If we make a = 10, in equation (1), we have the equation,

10x = n

(2)

If n is made equal to 1, in equation (2), the corresponding value of x is 0; if n is made equal to 10, the corresponding value of x is 1; if n is made equal to 100, the corresponding value of x is 2; and so on; hence, we have, from what precedes,

For all values of n between 1 and 10, the corresponding logarithms lie between 0 and 1; that is, they are fractions less than 1, and are generally expressed decimally. For all values of n between 10 and 100, the corresponding logarithms lie between 1 and 2; that is, they are equal to 1 plus a decimal. The logarithms

of all numbers between 100 and 1000, lie between 2 and 3; that is, they are equal to 2 plus a decimal.

In general, a logarithm is composed of two parts: an entire part, called the characteristic; and a decimal part, sometimes called the mantissa.

Logarithms are used to facilitate numerical computations, where they serve to convert operations of multiplication and division into the simpler ones of addition and subtraction. The following principles indicate the methods of applying logarithms to arithmetical computations.

Principles of Logarithms.

188. Let a denote the base of any system of logarithms, m and n any two numbers, and x and y their logarithms. We have, from equation (1),

multiplying (3) and (4), member by member, we have,

ax+y= mn;

whence, from the definition,

.x + y =

Log mn;

・ ・ (5)

hence, the following principle:

1o. The logarithm of the product of two numbers is equal to the sum of the logarithms of the two numbers.

If we divide (3) by (4), member by member, we have,

hence, the following principle:

2°. The logarithm of the quotient is equal to the logarithm of the dividend diminished by that of the divisor.

If we raise both members of (3) to any power denoted by p, we have,

apx = m2;

whence, by definition,

px = Log m2;

.

(7)

hence, the following principle:

3°. The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

If we extract any root of both members of (3), denoted by r, we have,

hence, the following principle:

4°. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root.

The applications of the above principles require a table of logarithms. A table of logarithms, is a table by means of which the logarithm corresponding to any number, or the number corresponding to any logarithm, may be found.

The principles above demonstrated, give rise to four practical

RULES.

1°. To find the product of two or more numbers:

Find the logarithms of the factors from a table, and take their sum; then find the number corresponding to the resulting logarithm, and it will be the product required.

2°. To find the quotient of one number by another:

Find the logarithms of the dividend and divisor from a table, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required.

3°. To raise a number to any power.

Find the logarithm of the number from a table, and multiply it by the exponent; then find the number corresponding to the resulting logarithm, and it will be the power required.

4. To extract any root of a number.

Find the logarithm of the number from a table, and divide it by the index; then find the number