Here the 2, to be carried, cancels the -2, and there remains the - to be set down. From the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the remainder will be the quotient required. NOTE. If I be to be carried to the index of the subtrahend, apply it according to the sign of the index; then change the sign of the index to -, if it be +, or to +, if it be ; and proceed according to the second note under the last rule. A Here 1, carried from the decimals to the -3, makes it become-2, which, taken from the other -2, leaves remaining. 4. To divide 7438 by 12'9476. Here the 1, taken from the -1, makes it become -2, to be set down. INVOLUTION BY LOGARITHMS. RULE. Multiply the logarithm of the given number by the index of the power, and the number answering to the product will be the power required. NOTE. A negative index, multiplied by an affirmative number, gives a negative product; and as the number carried from the decimal part is affirmative, their difference with the sign of the greater is, in that case, the index of the product. EXAMPLES. 4 Here 4 times the negative index being -8, and 3 to be carried, the difference -5 is the index of the product. To raise 10045 to the 365th root. EVOLUTION BY LOGARITHMS. RULE. Divide the logarithm of the given number by the index of the power, and the number answering to the quotient will be the root required. NOTE. When the index of the logarithm is negative, and cannot be divided by the divisor without a remainder, increase the index by a number, that will render it exactly divisible, and carry the units borrowed, as so many tens to the first decimal place; and divide the rest as usual. Here the divisor 2 is contained exactly once in the neg ative index -2, and therefore the index of the quotient is -1. Here the divisor 3 not being exactly contained in -4, 4 is augmented by 2, to make up 6, in which the divisor ' is contained just 2 times; then the 2, thus borrowed, being carried to the decimal figure 6, makes 26, which, di vided by 3, gives 8, &c. END OF LOGARITHMS. |