1:2 Find given line, i to tla given from fourt coli to be the le of o you. han rith. you des cas the. fi 1 06 a the characteristic only (16): hence, it appears, that the logarithms of tenths, are the same as the logarithms of the corresponding whole numbers, excepting, that the characteristic instead of being 0, is -1. If the fraction were of the form .06, it might be written ; taking the logarithms, log. =log. 06--2= 06 -2+log. 06. Now, the log 06 is but the log. 6; therefore, . the log. .06=2.778151, the minus sign belonging only to the characteristic, the decimal part being positive (16). If the decimal were.006, its logarithm would be the same, 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 place of figures; therefore, the logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to its logarithm a negative characteristic, greater by unity than the number of ciphers between the decimal point and the first significant place of figures. 19. To find, in the tables, a number answering to a given logarithm. Search, in the column of logarithms, for the decimal part of the given logarithm, and if it be exactly found, set down the corresponding number. Then, if the characteristic of the given logarithm be positive, point off, from the left of the number found, one more place of whole numbers than there are units in the characteristic of the given logarithm, and treat the other places as decimals: this will be the logarithm sought (9). If the characteristic of the given logarithm be 0, there will be one place of whole numbers ; if it be -1, the number will be entirely decimal; if it be -2, there will be one cipher between the decimal point and the first significant figure ; if it be -3, there will be two, &c. The number whose logarithm is 1.492481 is found in page 5, and is 31.08. But if the decimal part of the logarithm cannot be exactly found in the table, take the number answering to the next less logarithm ; take also from the table the corresponding difference in the column D: then, subtract this less logarithm from the given logarithm ; divide the remainder by the difference taken answering to the less logarithm: this gives the required number, nearly. This rule, like the one for finding the logarithm of a number when the places exceed four, supposes the numbers to be proportional to their corresponding logarithms. Ex. 1. To find the number answering to the logarithm 1.532708. Here, Next less log. is 1.532627, its number 34.09, the diff. 128. The difference between the given log. 1.532708 and 1.532627 is 81 ; therefore, 128) 8100 (63 which being decimals of a unit, in respect of the 9 in the number 34.09 must be annexed, and being so annexed, gives 34.0963 for the number answering to the log. 1.532708. Ex. 2. Required the number answering to the logarithm 3.233568. The given logarithm = 3.233568 The next less tabular logarithm of 1712 = 3.233504 . Diff. = 64 Tab. Diff. = 253) 64.00 (25 Hence the number sought is 1712.25, marking four places of integers for the characteristic 3. MULTIPLICATION BY LOGARITHMS. 20. If it be required to multiply numbers by means of their logarithms, we take from the table the logarithms of the numbers to be multiplied ; the sum of the logarithms is the logarithm of the product (4). The term sum is to be understood in its algebraical sense ; therefore, if any of the logarithms have negative characteristics, the difference between the sum of such characteristics, and the sum of the positive characteristics, is to be taken, and the sign of the greater prefixed. 1. To multiply 23.14 by 5.062. Log. 23,14 = 1.364363 2. To multiply 3.902, 597.16, and .0314728 together. Log. 3.902 0.591287 Log. 597.16 = 2.776091 Log..0314728 = 2.497935 Product, 73.3353 = 1.865313 Here, the 2 cancels the +2, and the I carried from the decimal part, is set down. 3. To multiply 3,586, 2.1046, 0.8372, and 0.0294, together. Log. 3.586 = 0.554610 Product, 0.1857618 = 1.268956 Here, the 2 carried cancels 2, and there remains ī to set down. DIVISION BY LOGARITHMS. : 21. If it be required to divide numbers by means of their logarithms, we have only to regard the dividend and divisor, as the numerator and denominator of a vulgar fraction; the logarithm of such fraction, found in the manner explained in articles 15 and 16, is the logarithm of the quotient. This additional caution may be added: that the difference of the logarithms, as there used, means the algebraical difference ; so that, if the logarithm of the denominator have a negative characteristic, its sign must be changed to positive, after adding to it the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of the numerator be negative, it must be treated as a negative number. 1. To divide 24163 by 4567. Log. 24163 = 4.383151 = 3.659631 Log. 4567 Quotient, 5.29078 = 0.723520 2. To divide .06314 by .007241. Log. .06314 2.800305 Log..007241 = 3.859799 Quotient, 8.71979 = 0.940506 Log. 37.119 = 1.569947 Quotient, .057447 = 2.759267 INVOLUTION BY LOGARITHMS. ..?. It it be required to raise a number to any power by means of logarithms, take from the table the logarithm of the number, and multiply it by the index of the power; the produet is the logarithm ot'the power: the number answering to thus logarithu is the power required. When the minus sign appears, it must be remembered, that it belongs to the characteristic only, and, therefore, the numbers which are carried for the tens from the decimal part of the kogarithm, are positive; hence, after the negative indices are multiplied by the intex of the root, such numbers must be subtracted trun the priuct, and the negative siga pretired, 1. To start the number 23791. Le 5791 = 0.411 408 lukler = |