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power, index and logarithms

LOGARITHM, to ,find the complement of a. Begin at the left hand, and write down what each figure wants of 9, only what the last significant figure wants of 10; so the complement of the logarithm of 456, viz. 2.6589648, is 7.3410352.

In the rule of three. Add the loga rithms of the second and third terms together, and from the sum subtract the logarithm of the first, the remainder is the logarithm of the fourth. Or, in stead of subtracting a logarithm, add its complement, and the result will be the same.

Looenirrots, to raise powers by. Mul tiply the logarithm of the number given by the index of the power required, the product will be the logarithm of the power sought.

Example. Let the cube of 32 be re quired by logarithms. The logarithm of 32 = 1.5051500, which, multiplied by 3, is 4.5151500, the logarithm of 32768, the cube of 32. But in raising powers, viz. squaring, cubing, &c. of any decimal fraction by logarithms, it must be observ ed, that the first significant figure of the power be put so many places below the place of units, as the index of its loga rithm wants of 10, 100, &c. multiplied by

the index of the power.

La GARS TIIMS, to extract the roots of powers by. Divide the logarithm of the number by the index of the power, the quotient is the logarithm of the root sought.

To „find mean proportionals between any two numbers. Subtract the logarithm of the least term from the logarithm of the greatest, and divide the remainder by a number more by one than the number of means desired ; then add the quotient to the logarithm of the least term (or subtract it from the logarithm of the great est) continually, and it will give the logarithms of all the mean proportionals required.

Example. Let three mean proportionals be sought, between 106 and 100.