logarithms of proper fractions ought to be negative, appears from the fact, that since a fraction expresses the quotient of the numerator divided by the denominator, applying the rule for division by logarithms, the greater logarithm would have to be subtracted from the lesser, and the remainder would of course be negative. From the above principles are derived the following rules: 1. To find the logarithm of a number consisting of significant figures with any number of ciphers annexed, find the logarithm of the significant figures, and make the characteristic one less than the number of figures in the given number including the ciphers. 2. To find the logarithm of a decimal or mixed number, consider the number as entire; find the decimal part of its logarithm, and make the characteristic one less than the number of figures in the entire part of the given number. 3. To find the logarithm of a decimal number having ciphers at the left; lock for the logarithm of the significant figures, and make the characteristic negative* and one more than the number of ciphers at the left of the given decimal. 51. CASE II.-We proceed now to the method of determining the logarithm of a number beyond the limits of the table. This method is by a simple calculation from the logarithms of numbers which the table contains, and depends upon the fact that the difference of any two numbers bears the same proportion to the difference of their logarithms, that the difference of two other numbers does to the difference of their logarithms, which is nearly true. (See Algebra, p. 284, note.) Take two numbers in the table differing from each other by 100, as the numbers 843700 and 843800, and a third number 843742 differing from the first of these by 42. The logarithm of the first number 843700, being the same as that of 8437, is given by the tables, and is 5.92619 The logarithm of the second number 843800 is 5.92624 Their difference is 5 * It is customary to write the negative sign over the characteristic, thus, 2.1756348. It affects the characteristic alone and not the decimal part of the logarithm, which must be considered as +. which may be found by subtraction, but to save this trouble the subtraction is performed, and the difference is written in the margin (in the tables of Callet in the right hand column marked dif.) Then on the principle that the difference of numbers is proportional to the difference of their logarithms, we have adding this to the logarithm of 843700 which is 5.92619 the sum, rejecting the two last places 10 which go beyond the usual number is which is the logarithm of 843742. ⚫0000210 5.92621 Had the first two numbers differed by 1000 instead of 100 the divisor in the value of a would have been 1000, and the quotient would have extended three places beyond the usual. Had they differed by 10, the quotient would have extended one place beyond. The inaccuracy of this method increases with the number of additional figures beyond four, in the number the logarithm of which is to be found.* From the above process may be observed the following rule: To find the logarithm of a number beyond the limits of the table. Enter the table with the first four figures of the given number, and find the corresponding logarithm. From the right hand margin take out the difference between this logarithm and the next in order in the table, and multiply it by the remaining figures of the proposed number, reject from the product as many figures to the right as there are in the multiplier, and add the rest of the product to the logarithm already found. EXAMPLE. 1. Required the logarithm of 739245. The decimal part of the log. of 7392 is 86876. *The same process applies to most other tables as well as tables of logarithms. In all tables the column corresponding to the column of numbers here is called the column of arguments. The other columns contain functions depending on these arguments. The number in the margin is Multiplying this by the remaining figures of the given number 6 45 Product, 270 From this product reject as many figures to the right as are contained in the multiplier, that is two in this case, and add the rest to the logarithm before found, namely, 86876 which is the decimal part of the log of 739245 required. Prefixing the proper characteristic, we have 5.86879. To facilitate the above calculations several smaller tables will be found in the right hand margin of the page, the use of which may be explained from an example. Suppose it be required to find the logarithm of 276738. After cutting off 38 on the right, I find the logarithm of the first four figures, 2767, to be 44201. The difference between this logarithm and the next is 16. In the left hand column of the little table in the margin headed 16, I find the first of the two figures cut off, 3, and against it the number 5, which I place under the logarithm already found, as seen in the scheme below. Again I find in the same way the second of the two figures cut off, 8, and against it 13, which I place as in the scheme below one place to the right. Adding these numbers taken from the little table to the first logarithm found, the sum 44207 is the logarithm sought. 44201 5 13 44207 or with the characteristic 5.44207 The right hand figure 3 of the 13, which would carry the decimal places beyond five, is rejected. If it were any figure greater than 5, we should add 1 to the last figure of the result, 7. The following example is worked with the tables of Callet in the same manner. * We add 1 for the rejected which is more than 100 The 7 neglected in adding up the above numbers being more than 5, or of the preceding place, 1 is added to that place. For the theory of the above see Algebra, at the end of Art. 214. PROBLEM II. To find the number corresponding to any given logarithm. 52. By referring to the proportion of Art. 51, and putting the value of x for the fourth term, we have Instead of the 42 being given and the 000021 required as before, the 000021 is now given and the 42 required. The first term of the proportion is 100 or 1000, &c., and the second term is diff. in the margin, to find the third term multiply the extremes and divide by the second term RULE. To find the number corresponding to any given logarithm. Seek for the decimal part of the given logarithm. If we find a logarithm exactly agreeing with that given, then the number, which the table shows us to belong to the logarithm found, will be the required number. If, however, as is most likely, we do not find the proposed logarithm exactly, then we are to take out the number corresponding to the next less logarithm; this number will of course fall short of that required, but the deficiency may be supplied as follows. Subtract the tabular logarithm from the given one, annex ciphers to the remainder at pleasure, and divide it by the diff. number in the margin, and annex the quotient to the number already taken from the table. N. B. Should there be a quotient figure without annexing a cipher to the dividend, this quotient figure must be added to the last figure of the number taken from the table. Should it be necessary to annex two ciphers before obtaining a quotient figure, a cipher must be placed in the quotient, and annexed with the figures that come after, to the number taken from the table. The logarithm next greater than that given may be taken from the tables, and the latter subtracted from the former, in which case you would subtract the quotient obtained by dividing the difference as above, instead of adding it. Annex ciphers to this diff. and divide by the diff. number in the margin, which is 6. 6)300 Annex the quotient 50 to the number 7392 before found, and you have the number required corresponding to the given logarithm, namely, 739250. This number contains six figures, one more than the characteristic of the given logarithm. In every case a sufficient number of ciphers must be annexed to obtain quotient figures enough, when appended, to make the whole of the number which thus results contain one more figure at least than is expressed by the characteristic of the given logarithm. If more quotient figures still be obtained, they will occupy the place of decimals. 2. Find the number of which the log. is 2.91345 Next less log. that of 8193 = 91344 5)10 2 53. To find the number corresponding to any given logarithm by the tables of Callet, seek the nearest logarithm in the tables, and subtract it from the given as directed above, then seek the remainder in the right hand column of the little table nearest, and if it be found, or a number not differing more than unity from it, the figure on the left of this number will express the sixth figure of the number required. The nearest number in the right hand column of the little table adjoining is 101, against which on the left is the figure 8, and the number sought is 345.678 |