63, from the line below, in the column headed 0, we have for the mantissa .63022. The reason for multiplying the tabular difference by .84 will be apparent from the following: log 1096005.03981. log 1097005.04021. The difference of the logarithms is 40 hundredthousandths, and the difference of the numbers is 100; but the difference of 109600 and 109684 is 84, which is .84 of 100; hence, the difference of the logarithms of 109600 and 109684 is .84 of 40 hundred-thousandths, which is 40 hundred-thousandths × .84 = 34 hundredthousandths, nearly. It is assumed that the difference of the logarithms of two numbers is proportional to the difference of the numbers, which is approximately true, especially if the numbers are large. 5. Find the logarithm of 123.613. The characteristic The mantissa of log 1236 Tab. diff. is 35; and 35×.13: log 123.613 2. .09202 5 2.09207 The tabular difference is .00035, and .00035 × .13 .0000455. But since the logarithms in this table are taken only to five decimal places, the two last figures, 55, are rejected, and 1 is carried to .00004, making .00005 for the correction. In general, when the left-hand figure of the part rejected exceeds 4, carry 1. When the tabular difference is large, as in the first part of the table, there may be small errors. Accordingly, for numbers between 10000 and 10900, it will be better to use the last two pages instead of the first page. 8. Rule. 1. If the number, or the product of the number by any power of 10, is found in the table, take the corresponding mantissa from the table, and prefix the proper characteristic. 2. If the number, without reference to the decimal point or O's on the right, is expressed by more than five figures, take from the table the mantissa corresponding to the first four or five figures on the left, multiply the corresponding tabular difference by the number expressed by the remaining figures, considered as a decimal, reject from the product as many figures on the right as are in the multiplier, carrying to the nearest unit, and add the result as so many hundredthousandths to the mantissa before found, and to the sum prefix the proper characteristic. 9. Examples. 1. What is the logarithm of 2347? Ans. 3.37051. 2. What is the logarithm of 108457? Ans. 5.03526. 3. What is the logarithm of 376542? 10. Problem. To find the number corresponding to a given logarithm. 1. What number corresponds to logarithm 2.03262? The mantissa is found in the column headed 8, and opposite 107 in the column headed N. Hence, without reference to the decimal point, the number corresponding is 1078; but since the characteristic is 2, the number is entirely decimal, and one 0 immediately follows the decimal point. Hence, the number corresponding is .01078. 2. What number corresponds to logarithm 2.83037? Since this logarithm can not be found in the table, take the next less, which is 2.83033, and the corresponding number, without reference to the decimal point, which is 6766. The difference between the given logarithm and the next less is 4, and the tabular difference is 6, which is the difference of the logarithms of the two numbers, 6766 and 6767, whose difference is 1. If the tabular difference of the logarithms, 6, corresponds to a difference in the numbers of 1, the difference of the logarithms, 4, will correspond to a difference of of 1; which, reduced to a decimal, and annexed to 6766, will give for the number, without reference to the decimal point, 676666. But since the characteristic is 2, there will be three integral places; hence, 676.666 is the number required. 3. What number corresponds to logarithm 2.76398? The given log = 2.76398 ... number: Next less log = 2.76395 580.737 .'. number € 580.7 It is necessary to write only that part of the next less logarithm which differs from the given logarithm. Conceive O's annexed to the difference, and divide by the tabular difference; and annex the quotient to the number corresponding to the next less logarithm. In practical work abbreviate thus: Let denote the given logarithm; l', the next less logarithm; n and n', the corresponding numbers; t, the tabular difference; d, difference of logarithms; c, the correction. 4. What number corresponds to logarithm 1.73048? l=1.73048 ... n .537625 l'= 1.73046 ... n'= .5376 1. If the given mantissa can be found in the table, take the number corresponding, and place the decimal point according to the law for the characteristic. 2. If the given mantissa can not be found in the table, take the next less and the corresponding number. Subtract this mantissa from the given mantissa, annex O's to the remainder, divide the result by the tabular difference, annex the quotient to the number corresponding to the logarithm next less than the given logarithm, and place the decimal point according to the law for the characteristic. 4. What number corresponds to logarithm 1.48237? Ans. .30365. 5. What number corresponds to logarithm 3.63025? Ans. .0042683. MULTIPLICATION BY LOGARITHMS. 13. Proposition. The logarithm of the product of two numbers is equal to the sum of their logarithms. y m; then, by def., log m = x. n; then, by def., log n = y. (1)×(2)=(3) b**=mn; then, by def., log mn=x+y. ... log mn log m+log n. 14. Rule. 1. Find the logarithms of the factors and take their sum, which will be the logarithm of the product. 2. Find the number corresponding which will be their product. |