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ART. 1. The operations of Multiplication and Division, when they are to be often repeated, become so laborious, that it is an object of importance to substitute, in their stead, more simple methods of calculation, such as Addition and Subtraction. If these can be made to perform, in an expeditious manner, the office of multiplication and division, a great portion of the time and labor which the latter processes require, may be saved.

Now it has been shown, (Algebra, 189, 193,) that powers may be multiplied by adding their exponents, and divided, by subtracting their exponents. In the same manner, roots may be multiplied and divided, by adding and subtracting their fractional exponents. (Alg., 232, 239.) When these exponents are arranged in tables, and applied to the general purposes of calculation, they are called Logarithms.



In forming a system of logarithms, some particular number is fixed upon, as the base, radix, or first power, whose logarithm is always 1. From this a series of powers is raised, and the exponents of these are arranged in tables for use. To explain this, let the number which is chosen for the

first power be represented by a. Then taking a series of powers, both direct and reciprocal, as in Alg. 163;

a*, a3, a3, a1, ao, a ̄ ̄1, a ̄ ̄2, a ̄ ̄3, a ̄*, &c.

The logarithm of a3 is 3, and the logarithm of a1 is-1,

of a1 is 1,

of ao is 0,

of a is-2,

of a

is-3, &c.

Universally, the logarithm of a* is x.

3. In the system of logarithms in common use, called Briggs's logarithms, the number which is taken for the radix or base is 10. The above series, then, by substituting 10 for a, becomes

10, 10, 10, 10', 10°, 10, 10, 10", &c. Or 10000, 1000, 100, 10, 1, to, Tou, Tobo, &c.

Whose logarithms are

4, 3, 2, 1, 0, -1, −2, -3, &c.

4. The fractional exponents of roots, and of powers of roots, are converted into decimals, before they are inserted in the logarithmic tables. See Alg. 208.

The logarithm of a3, or a°.****, is 0.3333,

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These decimals are carried to a greater or less number of places, according to the degree of accuracy required.

5. In forming a system of logarithms, it is necessary to obtain the logarithm of each of the numbers in the natural series 1, 2, 3, 4, 5, &c.; so that the logarithm of any number may be found in the tables. For this purpose, the radix of the system must first be determined and then every other number may be considered as some power or root of


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