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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, n the number whose logarithm is considered, and ±x, 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 is 10; and this system affords the greatest facilities in calculations, because 10 is the base of the common numeration, both in whole numbers and decimal fractions.
4. Taking a±2=n, we have, ±x=log. n; and putting a±= m, gives, iy=log. m. If the equations, a2=n, and aa=m, be multiplied together, member by member, we have, az Xay= nXm, or a2+y=nXm. In this expression, x+y is the logarithm of n×m (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" a*=n, a=m, be divided, member by ; or at-y-. In this expression, x-y is the
logarithm of (2); from which we conclude, that the differ
ence of the logarithms of any two numbers, is equal to the logarithm of their quotient.