mber whose logarithm is taken. first case, for numbers between 1 and 10, place of figures, and the characteristic is 0. etween 10 and 100, there are two places of e characteristic is 1; and similarly for other TABLE OF LOGARITHMS. f logarithms, is a table in which are written of all numbers between 1 and some other given = logarithms of all numbers between 1 and itten in the annexed table. column on the left of each page of the table, of numbers, and is designated by the letter N; s of these numbers are placed directly opposite the same horizontal line. from the table, the logarithm of any number. number is less than 100, look on the first page of ng the column of numbers under N, until the und: the number directly opposite, in the column og, is the logarithm sought. Thus, log 9=0.954243. number is greater than 100, and less than 10,000. the characteristic of every logarithm is less by the places of integer figures in its corresponding Tt. 4), its value is known by a simple inspection ber whose logarithm is sought. Hence, it has not ed necessary to write the characteristics in the table. In the decimal part of the logarithm, find, in the mumbers, the first three figures of the given number. s across the page, along a horizontal line, into the marked 0, 1, 2, 3, 4, 5, &c., until you come to the hich is designated by the fourth figure of the given at this place there are four figures of the required - To the four figures so found, two figures taken column marked o, are to be prefixed. If the four Lus found, stand opposite to a row of six figures in the marked 0, the two figures from this column, which the four figures found are opposite a line of only four figure you are then to ascend the column till you come to the lin of six figures; the two figures, at the left hand, are to b prefixed, and then the decimal part of the logarithm is ol tained; to which prefix the characteristic, and you have th entire logarithm sought. For example, log 1122=3.049993 log 89793.953228 In several of the columns, designated 0, 1, 2, 3, 4, &c., sma dots are found. When the logarithm falls at such places a cipher must be written for each of the dots, and the tw figures, from the column 0, which are to be prefixed, are ther found in the horizontal line directly below. the two dots being changed into two ciphers, and the 34 to be taken from the column 0, is found in the horizontal line directly below. The two figures from the column 0, must also be taken from the horizontal line below, if any dots shall have been passen over, in passing along the horizontal line: thus, log 30983.491081 the 49 from the column 0, being taken from the line 310. When the number exceeds 10,000, or is expressed by five or more figures. 9. Consider all the figures, after the fourth from the left hand, as ciphers. Find from the table the logarithm of the first four figures, and to it prefix a characteristic less by unity than all the places of figures in the given number. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the figures that have been considered as ciphers: then cut off from the right hand as many places for decimals as there are figures in the multiplier, and add the product so obtained to the first logarithm, and the sum will be the logarithm sought. Let it be required, for example, to find the logarithm of 672887. log 672800=5.827886 the characteristic being 5, since there are six places of figures. 5; then pointing off two decimal places, we the number to be added. log 672800=5.827886 +56.55 log 672887=5.827943. proportional number, we omit the decimal the decimal part exceeds 5 tenths, as in the value is nearer unity than 0; in which case, yone, the figure on the left of the decimal thod of finding the logarithms of numbers our places of figures, does not give the exact it supposes that the logarithms are proporcorresponding numbers, which is not rigorously he reason of the above method, let us take the 72900, a number greater than 672800 by 100. 65=difference of loga 100: 65:: 87: 56.55 portion the first term 100 is the difference beumbers, one of which is greater and the other given number; and the second term 65 is the their logarithms, or tabular difference. term 87 is the difference between the given numLess number 672800; and hence the fourth term difference of their logarithms. This difference ded to the logarithm of the less number, will give greater, nearly. been three figures of the given number treated The first term would have been 1000; had there it would have been 10000, &c. Therefore, the e rule, for the use of the column of differences, is To find the logarithm of a decimal number. and a decimal, such as 36.78, it may be put under the fo 3678 7. But since a fraction is equal to the quotient obtai by dividing the numerator by the denominator, its logarit will be equal to the logarithm of the numerator minus 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 trea as though it were entire, except in fixing the value of characteristic, which is always one less than the number of integer figures. 12. The logarithm of a decimal fraction is also rea found. For, log 0.8=log=log 8-1=-1+log 8. But, 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 chara ristic. Hence it appears, that the logarithm of tenths is same as the logarithm of the corresponding whole num 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, log %%=log 06-2=-2+log 06=-2.778151 in which the minus sign, as before, belongs only to the ch acteristic. If the decimal were 0.006 its logarithm would the same as before, excepting the characteristic, which wo be-3. It is, indeed, evident, that the negative characteri will always be one greater than the number of ciphers tween the decimal point and the first significant figu Therefore, the logarithm of a decimal fraction is found, considering it as a whole number, and then prefixing to the d mal part of its logarithm a negative characteristic greater unity than the number of ciphers between the decimal point the first significant figure. That we may not, for a moment, suppose the negative s to belong to the whole logarithm, when in fact it belongs o to the characteristic, we place the sign above the charac ristic, thus, table, the number answering to a given logarithm. in the columns of logarithms for the decimal ven logarithm, and if it can be exactly found, orresponding number. Then, if the characterven logarithm is positive, point off from the left - found, one more place for whole numbers than s in the characteristic of the given logarithm, figures to the right as decimals. acteristic of the given logarithm is 0, there will of whole numbers; if it is -1, the number will Lecimal; if it is -2, there will be one cipher decimal point and the first significant figure ; here will be two, &c per whose logarithm is 1.492481, is found at page 08. the decimal part of the logarithm cannot be d in the table, take the number answering to the logarithm; take also from the table the corresference in the column D. Then, subtract this am from the given logarithm, and having annexed of ciphers to the remainder, divide it by the difen from the column D, and annex the quotient to answering to the less logarithm: this gives the umber, nearly. This rule, like that for finding thm of a number when the places of figures exsupposes the numbers to be proportional to their Ling logarithms. the number answering to the logarithm 1.532708. logarithm is • 1.532708 1.532627 81 |