being multiplied by 87, the figures regarded as ciphers, gives for a product 5655; then pointing off two decimal places, we obtain 56.55 for the number to be added. Hence Adding gives • log 672800=5.827886 +56.55 log 672887=5.827943. In adding the proportional number, we omit the decimal part; but when the decimal part exceeds 5 tenths, as in the case above, its value is nearer unity than 0; in which case, we augment by one, the figure on the left of the decimal point. 10. This method of finding the logarithms of numbers which exceed four places of figures, does not give the exact logarithm; for, it supposes that the logarithms are proportional to their corresponding numbers, which is not rigorously true. To explain the reason of the above method, let us take the logarithm of 672900, a number greater than 672800 by 100. We then have, In this proportion the first term 100 is the difference between two numbers, one of which is greater and the other less than the given number; and the second term 65 is the difference of their logarithms, or tabular difference. The third term 87 is the difference between the given number and the less number 672800; and hence the fourth term 56.55 is the difference of their logarithms. This difference therefore, added to the logarithm of the less number, will give that of the greater, nearly. Had there been three figures of the given number treated as ciphers, the first term would have been 1000; had there been four, it would have been 10000, &c. Therefore, the reason of the rule, for the use of the column of differences, is manifest. To find the logarithm of a decimal number. 3678 이이이 and a decimal, such as 36.78, it may be put under the form But since a fraction is equal to the quotient obtained by dividing the numerator by the denominator, its logarithm will be equal to the logarithm of the numerator minus the logarithm of the denominator. Therefore, log=log 3678-log 100=3.565612-2=1.565612 from which we see, that a mixed number may be treated as though it were entire, except in fixing the value of the characteristic, which is always one less than the number of the integer figures. 12. The logarithm of a decimal fraction is also readily found. For, log 0.8=log&log 8-1=-1+log 8. But, log 8=0.903090 which is positive and less than 1. Therefore, log 0.8=-1+0,903090=-1.903090 in which, however, the minus sign belongs only to the characteristic. Hence it appears, that the logarithm of tenths is the same as the logarithm of the corresponding whole number, excepting, that the characteristic instead of being 0, is-1. If the fraction were of the form 0.06 it might be written; taking the logarithms, we have, logo=log 06-2=-2+log 06=-2.778151 in which the minus sign, as before, belongs only to the characteristic. If the decimal were 0.006 its logarithm would be the same as before, excepting the characteristic, which would be-3. It is, indeed, evident, that the negative characteristic will always be one greater than the number of ciphers between the decimal point and the first significant figure. Therefore, the logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to the decimal part of its logarithm a negative characteristic greater by unity than the number of ciphers between the decimal point ana the first significant figure. That we may not, for a moment, suppose the negative sign to belong to the whole logarithm, when in fact it belongs only to the characteristic, we place the sign above the characteristic, thus, To find in the table, the number answering to a given logarithm. 13. Search in the columns of logarithms for the decimal part of the given logarithm, and if it can be exactly found, set down the corresponding number. Then, if the characteristic of the given logarithm is positive, point off from the left of the number found, one more place for whole numbers than there are units in the characteristic of the given logarithm, and treat the figures to the right as decimals. If the characteristic of the given logarithm is 0, there will be one place of whole numbers; if it is -1, the number will be entirely decimal; if it is -2, there will be one cipher between the decimal point and the first significant figure ; if it is-3, there will be two, &c The number whose logarithm is 1.492481, is found at page 5, and is 31.08. But when the decimal part of the logarithm cannot be exactly found in the table, take the number answering to the nearest less logarithm; take also from the table the corresponding difference in the column D. Then, subtract this less logarithm from the given logarithm, and having annexed any number of ciphers to the remainder, divide it by the difference taken from the column D, and annex the quotient to the number answering to the less logarithm: this gives the required number, nearly. This rule, like that for finding the logarithm of a number when the places of figures exceed four, supposes the numbers to be proportional to their corresponding logarithms. 1. Find the number answering to the logarithm 1.532708. The 63 being annexed to the tabular number 34.09 gives 34.0963 for the number answering to the logarithm 1.532708. 2. Required the number answering to the logarithm Hence the number sought, is 1712.25, marking four places for integers since the characteristic is 3. MULTIPLICATION BY LOGARITHMS. 14. When it is required to multiply numbers by means of their logarithms, we first find from the table the logarithms of the numbers to be multiplied; we next add these logarithms together, and their sum is the logarithm of the product of the numbers (Art. 2). The term sum is to be understood in its algebraic sense; therefore, if any of the logarithms have negative characteristics, the difference between their sum and that of the positive characteristics, is to be taken, and the sign of the greater prefixed. Here the cancels the +2, and the 1 carried from the 3. Multiply 3.586, 2.1046, 0.8372, and 0.0294, together. In this example the 2, carried from the decimal part, can cels 2, and there remains i to be set down. DIVISION OF NUMBERS BY LOGARITHMS. 15. When it is required to divide numbers by means of their logarithms, we have only to recollect, that the subtraction of logarithms corresponds to the division of their numbers (Art. 3). Hence, if we find the logarithm of the dividend, and from it subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient. This additional caution may be added. The difference of the logarithms, as here used, means the algebraic difference; so that, if the logarithm of the divisor have a negative characteristic its sign must be changed to positive, after diminishing it by the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of the dividend is negative, it must be treated as a negative number. Here, 1 carried from the decimal part to the 3 changes it to 2, which being taken from 2, leaves o for the characteristic. 3. To divide 37.149 by 523.76 log 37.149=1.569947 |