or six figures are required, and divide the number thus produced, by the former difference; annex the quotient to the right hand of the four figures already found, and it will give the natural number required. Thus let it be required to find the natural number corresponding to the logarithm 2.53899 true to five figures; then, Divide 40 by 13 and the quotient will be 3, which, annexed to the right hand of 3459, the four figures already found, makes 34593; therefore as the index is 2, the required natural number is 345.93. Again let it be required to find the natural number corresponding to the logarithm 4.59859, true to six figures; then, Divide 200 by 11, and the quotient will be 18, which ready found, makes 396818; therefore as the index is 4, the required natural number is 39681.8. EXAMPLES. 1. Required the natural number answering to the logarithm 1.88030. Ans. 75.91. 2. Required the natural number answering to the logarithm 5.37081. Ans. 234861. 3. Required the natural number answering to the logarithm 3.11977. Ans. 1317.56. 4. Required the natural number answering to the logarithm-2.97435. Ans. .094265. PROBLEM III. To multiply numbers by means of logarithms. Case 1.-When all the factors are whole or mixed numbers. RULE. Add together the logarithms of the factors, and the sum will be the logarithm of the product. 34. 2. Required the continued product of 17.3, 1.907 and 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 .003179 is -3.50229 Product .000201 Sum -4.30319 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. Logarithm of 3450 is 3.53782 Do. 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 Ans. 21.3585. 1.73958. 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 name with that of the dividend, add the indices together |