part of the logarithms, which added to 3, the sum of the affirmative indices, makes 5; from this taking 4, the sum of the negative indices, the remainder is 1, which is the index of the sum of the logarithms, and is affirmative, because the sum of the affirmative indices together with the number carried, exceeds the sum of the negative indices. 2. Required the continued product of .0839, .7536, and .003179. In this example there is 2 to carry from the decimal part of the logarithms, which subtracted from 6, the sum of the negative indices, leaves 4, which is the index of the sum of the logarithms, and is negative, because the sum of the negative indices is the greater. 3. Required the continued product of 13.19, .3765, and .00415. Ans. .02061. 4. Required the continued product of 343, 1.794, 5.41 and .019. Ans. 63.25. PROBLEM IV. To divide numbers by means of Logarithms. CASE 1. When the dividend and divisor are both whole RULE. From the logarithm of the dividend, subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient. Note.-When the divisor exceeds the dividend, the question must be wrought by the rule given in the next case. EXAMPLES. 1. Required the quotient of 3450 divided by 23. 2. Required the quotient of 420.75 divided by 76.5. Ans. 5.5. 3. Required the quotient of 37.1542 divided by 1.73958. Ans. 21.3585. CASE 2. When the dividend or divisor, or both of them, are decimal numbers. RULE. Subtract the decimal parts of the logarithms as before, and if 1 be borrowed in the left hand place of the decimal part, add it to the index of the divisor when that index is affirmative, but subtract it when negative. Then conceive the sign of the index of the divisor changed from affirmative to negative, or from negative to affirmative; and if, when changed, it be of the same but if it be of a different name, take the difference of the indices and prefix the sign of the greater. EXAMPLES. 1. Required the quotient of .7591 divided by 32.147. In this example the index of the divisor, with its sign changed, is-1, which added to -1, the index of the dividend, makes-2, for the index of the quotient. 2. Required the quotient of .63153 divided by .00917. In this example there is 1 to carry from the decimal part of the logarithm, which subtracted from -3, the index of the divisor, leaves -2; this with its sign changed is +2; from which subtracting 1 the index of dividend, the remainder is 1, and is affirmative because the affirmative index is the greater. 3. Required the quotient of 13.921 divided by 7965.13. Logarithm of 13 921 is 1.14367 Do. 7965.13 is 3.90125 In this example there is 1 to carry from the decimal part of the logarithm, which added to 3, the index of the divisor, makes 4; this with its sign changed is -4; from which subtracting 1, the index of the dividend, the remainder is -3. 4. Required the quotient of 79.35 divided by .05178. Ans. 1532.46. 5. Required the quotient of .5903 divided by .931. Ans. .63404. PROBLEM V. To involve a number to any power; that is to find the square, cube, &c. of a number, logarithmically. RULE. Multiply the logarithm of the given number by the index of the power, viz. by 2 for the square, by 3 for the cube, &c. and the product will be the logarithm of the power. Note. When the index of the logarithm is negative, if there be any to carry from the decimal part, instead of adding it to the product of the index and multiplier, subtract it, and the remainder will be the index of the logarithm of the power, and will always be negative. EXAMPLES. 1. Required the square of 817. Logarithm of 317 is 2.50106 2 4. Required the cube of 7.503. Ans. 422.37. 5. Required the 7th power of .32513. Ans. .0003841. PROBLEM VI. To extract any root of a number logarithmically. RULE. Divide the logarithm of the given number by the index of the root, that is by 2 for the square root, by 3 for the cube root, &c. and the quotient will be the logarithm of the required root. Note. When the index of the logarithm is negative, and does not exactly contain the divisor, increase the index by a number just sufficient to make it exactly divisible by it, and carry the units borrowed, as so many tens, to the left hand figure of the decimal part; then |