The natural number consisting of five integers, the index must be 4; therefore the log. of 59686 is 4.77587. Again, let it be required to find the log. of .0131755; then, As the given number is a decimal, and has one cipher between the decimal point and first significant figure, the index must be -2; therefore the log. of .0131755 is -2.11977. PROBLEM II. logarithm. If four figures only be required in the answer, look in the table for the decimal part of the given logarithm, and if it cannot be found exactly, take the one nearest to it, whether greater or less; then the three figures in the first column, marked No, which are in a line with the logarithm found, together with the figure at the top of the table directly above it, will form the number required. Observing, that when the index of the given lo. garithm is affirmative, the integers in the number found must be one more than the number expressed by the index; but when the index of the given logarithm is nega. tive, the number found will be wholly a decimal, and must have one cipher less placed between the decimal point and first significant figure on the left hand, than the number expressed by the index. Thus the natural number corresponding to the logarithm 2.90233 is 798.6, the natural number corresponding to the logarithm 3.77055 is 5896, and the natural number corresponding to the logarithm --3.36361 is .00231. If the exact logarithm be found in the table, and the figures in the number corresponding do not exceed the index by one, annex ciphers to the right hand till they do. Thus the natural number corresponding to the loga. rithm 6.64068 is 4372000. If five or six figures be required in the answer, find, in the table, the logarithm next less than the given one, and take out the four figures answering to it as before. Subtract this logarithm from the next greater in the table, and also from the given logarithm; to the latter dif. 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, Given logarithm ...53899 .53895 the natural number corresponding is 3459 Diff. with one cipher annexed 40 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 fi. gures; then, Given logarithm .59859 59857, the natural number an swering to it is 3968. Diff. with two ciphers annexed 200. Divide 200 by 11, and the quotient will be 18, which annexed to the right hand of 3968, the four figures already found, makes 396818; therefore as the index is 4, the re. quired natural number is 39681.8. EXAMPLES. 1. Required the natural number answering to the lo. garithm 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. EXAMPLES. 1. Required the product of 84 by 56. Logarithm of 84 is 1.92428 Do. of 56 is 1.74819 Product 4904 Sum 3.67247 2. Required the continued product of 17.3, 1.907 and 34. Logarithm of 17.3 is 1.23805 1.907 is 0.28035 34. is 1.53148 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 deci. mal 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 decima C |