University Algebra

called the common system; and that system whose base is 2.718281828... called the Naperian system.

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

184. If we make a = 10, in Equation (1), we shall have the equation,

When n is made equal to 1, in Equation (2), the corresponding value of x is 0. When n is made equal to 10, the corresponding value of x is 1. When 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 cor responding 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 often used to facilitate numerical com putations, where they serve to convert operations of mul

tiplication and division into the simpler ones of addition and subtraction. The following principles indicate the methods of applying logarithms to arithmetical compu. cations.

GENERAL PRINCIPLES.

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

Multiplying (3) and (4), 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 two numbers.

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

That is, 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,

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m2.

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.

If we denoted by r, we have,

extract any root of both members of (3),

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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 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, ani 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 loga rithm, 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 corresponding to the resulting logarithm, and it will be the root required.

No practical examples can be given to illustrate the preceding rules, without a table of logarithms. A few examples of transformation are, however, annexed, which serve to show the methods of proceeding in the em ployment of logarithms.