3. Find by logarithms the product of 76.5 by 5.5 Ans. 420.75. 4. Find by logarithms the continued product of 42.35, 1.7364, and 1.76. Ans. 129.424. CASE 2.-When some or all of the factors are decimal numbers. RULE. Add the decimal parts of the logarithms as before, and if there be any to carry from the decimal part, add it to the affirmative index or indices, or else subtract it from the negative. Then add the indices together, when they are all of the same kind; that is, all affirmative or all negative; but when they are of different kinds, take the difference between the sums of the affirmative and negative ones, and prefix the sign of the greater. Note.—When the index is affirmative, it is not necessary to place any sign before it; but when it is negative, the sign must not be omitted. EXAMPLES. 1. Required the continued product of 349.17, 25.43, 93521 and .00576. In this example there is 2 to carry from the decimal 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. Logarithm of .0839 is -2.92376 Do. .7536 is -1.87714 Product .000201 Sum -1.30319 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 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. 23 is 1.36173 Quotient 150 Remainder 2.17609 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 i 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 name with that of the dividend, add the indices together 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 Logarithm of .7591 is -1.88030 Do. 32.147 is 1.50714 Quotient .02361 Remain. -2.37316 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. Logarithm of 63153 is .00917 is -1.80039 Quotient 68.8683 Remain. 1.83802. 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 the dividend, the remainder is 1, and is affirmative, be cause the affirmative index is the greater. 3. Required the quotient of 13.921 divided by 7965.13 Logarithm of Do. 13.921 is 1.14367 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, fc. 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. Rcquired the square of 317. Logarithm of 317 is 2.50106 |