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1. LOGARITHMS of numbers are the indices that denote the different powers to which a given number must be raised to produce those numbers.

2. If a be the given number, whose indices and powers are to be considered, then a +* being put equal to n, a, the given

number, or root, is called the base of the system of logarithms, in the number whose logarithm is considered, and tr, the logarithm of that number.

3. Any number, except 1, may be taken for the base of a system of logarithms. In the system in general use, the base x is 10; and this system affords the greatest facilities in calcula

tions, because 10 is the base of the common numeration, both in whole numbers and decimal fractions.

4. Taking a£=n, we have, Ex==log. n; and putting a+= m, gives, iy=log. m. If the equations, ar=n, and ay=m, be multiplied together, member by member, we have, ar X ay = nxm, or qzty=n X m. In this expression, x+y is the logarithm of nxm (2); from which we conclude, that the sum of the logarithms of any two numbers, is equal to the logarithm of their product.

5. If the equations ař=n, ah=m, be divided, member by meniber, ; or al-43-. In this expression, *—y is the ay

m logarithm of " (2); from which we conclude, that the difference of the logarithms of any two numbers, is equal to the log«. rithm of their quotient.

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