RULE.-The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of o's that immediately follow the decimal point. The characteristic alone is negative, the mantissa being always positive. This fact is indicated by writing the negative sign over the characteristic:"thus, 2.371465, is equivalent to 2 + .371465. NOTE.—It is to be observed, that the characteristic of a mixed number is the same as that of its entire part. Thus, the mixed number 74.103 lies between 10 and 100; hence, its logarithm lies between 1 and 2, as does the logarithm of 74. GENERAL PRINCIPLES. 5. Let m and n denote any two numbers, and p and a their logarithms. We shall have, from the definition of a logarithm, the following equations, Multiplying (4) and (5), member by member, we have, 10P +9 = mn; whence, by the definition, p + q = log (mn). (6.) That is, the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 6. Dividing (4) by (5), member by member, we have, m 10P-4 = n whence, by the definition, p - 9 т log (m). . (7.) n That is, the logarithm of a quotient is equal to the logarithm of the dividend diminished by that of the divisor. 7. Raising both members of (4), to a power denoted by t, we have, 10pt = m'; whence, by the definition, (8.) That is, the logarithm of any power of a number, is equal to the logarithm of the number multiplied by the exponent of the power. 8. Extracting the root, indicated by r, of both members of (4), we have, 107 ♡ whence, by the definition, m; That is, the logarithm of any root of a number, is equal to the logarithm of the number divided by the index of the root. The preceding principles enable us to abbreviate the operations of multiplication and division of numbers, by the addition and subtraction of their logarithms. TABLE OF LOGARITHMS. 9. A TABLE OF LOGARITHMS, is a table by means of which we can find the logarithm corresponding to any number, or the number corresponding to any logarithm. In the table appended, the complete logarithm is given for all numbers from 1 up to 100. For other numbers, between 100 and 10,000, the mantissas alone are given; the characteristic may be found by one of the rules of (Art. 4). Before explaining the uses of the table, it is to be shown that the mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact power of 10. Let n denote any number whatever, and 10o any power of 10, p being any whole number, either positive or negative. Then, in accordance with the principles of (Arts. 5 and 3), we shall have, log (n X 10P) = log n + log 10% = p + log n; but p is, by hypothesis, a whole number; hence, the decimal part of the log (n X 10P), is the same as that of log n; which was to be proved. Hence, in finding the mantissa of the logarithm of a number, we may regard the number as a decimal, and move the decimal point to the right or left, at pleasure. Thus, the mantissa of the logarithm of 456357, is the same as that of the number 4563.57; and the mantissa of the logarithm of 2.00357, is the same as that of 2003.57. 1'. To find, from the table, the logarithm of a number less than 100. 10. Look on the first page, in the column headed “N," for the given number; the number opposite is the logarithm required. Thus, log 67 = 1.826075. 2. To find the logarithm of a number between 100 and 10,000. 11. Find the characteristic by the first rule of (Art. 4). To find the mantissa, look in the column headed “N," for the first three figures of the number; then pass along a horizontal line until you come to the column headed with the fourth figure of the number; at this place will be found four figures of the mantissa, to which, two other figures, taken from the column headed “0,” are to be prefixed. If the figures found stand opposite a row of six figures, in the column headed “0,” the first two of this row are the ones to be prefixed; if not, ascend the column till a row of six figures is found; the first two, of this row, are the ones to be prefixed. If, however, in passing back from the four figures, first found, any dots are passed, the two figures to be prefixed must be taken from the line immediately below. When the figures first found, fall at a place where dots occur, the dots must be replaced by O's, and the figures to be prefixed must be taken from the line below. Thus, Log 8979 = 3.953228 3°. To find the logarithm of a number greater than 10,000. 12. Find the characteristic by the first rule of (Art. 4). To find the mantissa, place a decimal point after the fourth figure (Art. 9), thus converting the number into a mixed number. Find the mantissa of the entire part, by the method last given. Then take from the column headed “D,” the corresponding tabular difference, and multiply this by the decimal part and add the product to the mantissa just found. The result will be the roquired mantissa. It is to be observed that when the decimal part of the product just spoken of is equal to or exceeds .5, we add i to the entire part; otherwise the decimal part is rejected. EXAMPLE. To find the logarithm of 672887. The characteristic is 5. Placing a decimal point after the fourth figure, the number becomes 6728.87. The mantissa of the logarithm of 6728 is 827886, and the corresponding number in the column “D," is 65. Multiplying 65 by .87, we have 56.55; or, since the decimal part exceeds .5, 57. We add 57 to the mantissa already found, giving 827943, and we finally have, log 672887 = 5.827943. The numbers in the column “D” are the differences between the logarithms of two consecutive whole numbers, and are found by subtracting the number under the heading “4” from that under the heading “5." In the example last given, the mantissa of the logarithm of 6728 is 827886, and that of 6729 is 827951, and their difference is 65; 87 hundredths of this difference is 57: hence, the mantissa of the logarithm of 6728.87, is found by adding 57 to 827886. The principle employed is, that the differences of numbers are proportional to the differences of their logarithms, when these differences are small. 4°. To find the logarithm of a decimal 13. Find the characteristic by the second rule of (Art. 4). To find the mantissa, drop the decimal point, and thus consider the decimal a whole number. Find the mantissa of |