13. Hence, by means of logarithms, we can perform multiplication by addition, and division by subtraction; also, we can raise a number to any power by a single multiplication, and extract any root of a number by a single division. 14. All numbers, integral, fractional, or mixed, having the same succession of significant figures, have logarithms with the same decimal part. For since the logarithm of 10 is 1, the product of any number by 10 will have a logarithm increased by 1; and, likewise, the quotient of any number divided by 10 will have a logarithm diminished by 1; and, 1 being an integer, the logarithms will differ only in their characteristics. 15. The negative sign placed over the characteristic indicates that it alone is negative, the decimal part being always positive. TABLE OF LOGARITHMS. 16. A Table of Logarithms usually contains all the whole numbers between 1 and a given number, with their logarithms. The accompanying table contains the logarithms of all numbers from 1 up to 10,000, calculated to six places of decimals. 17. In the table, the characteristics of the logarithms of the first 100 numbers are inserted; but for all other numbers the decimal part only of the logarithms is given, while the characteristic is left to be supplied by inspection, according to the principles already furnished (Art. 7, 8). 18. The numbers are in the column headed N, and their logarithms, or the decimal parts of their logarithms, are opposite on the same line. When the first two figures of the decimal are the same for several successive logarithms, they are not repeated for each, but, being used once, are then left to be supplied. 19. In the column headed D are the mean or average differences of the ten logarithms against which they are placed. TO FIND THE LOGARITHM OF ANY NUMber. 20. When the given number is any integer of ONE or TWO figures. Look on the first page of the table, and opposite the given number will be found the logarithm with its characteristic. Thus, the logarithm of 63 is 1.799341; 21. When the given number is any integer of THREE FIG URES. Look in the table for the given number, and opposite the same, in the column headed 0, will be found the decimal part of the logarithm, to which must be prefixed 2 as the characteristic (Art. 7). Thus, the logarithm of 110 is 2.041393; 22. When the given number is any integer of FOUR figures, either with or without ciphers annexed. Find the first three figures of the given number in the column headed N, and, opposite to them, in the column headed by the fourth figure, will be found the decimal part of the logarithm; to which the characteristic, as determined by Art. 7, must be prefixed. Thus, the logarithm of 4901 is 3.690285; 23. When the given number is any integer of FIVE or MORE figures. Find the logarithm of the first four figures as in Art. 22, regarding the others as ciphers annexed; then take, from the column headed D, the number on the same horizontal line with the decimal part of the logarithm, and multiply it by the remaining figures of the given number; reject from the right of the product thus obtained as many figures as there were in the multiplier, and add what is left to the decimal part of the logarithm already found; and the sum will be the required logarithm. it be required to find the logarithm of 93192: Thus, if the logarithm of 93190 is 4.969369 Dif. from col. D, 47 This process is based upon the supposition that the differences of logarithms are proportional to the differences of their corresponding numbers, which is not strictly correct, yet sufficiently exact for practical purposes. When the figure or figures rejected from the right of the product, considered decimally, exceed in value .5, the right-hand figure of what is left to be added must be increased by 1, to insure greater accuracy in the result. 24. When the given number is any DECIMAL FRACTION, or any mixed number expressed decimally. Find the decimal part of the logarithm of the number, as if it were an integer, and prefix the proper characteristic (Arts. 7 and 8). Thus, the logarithm of 93.192 is 1.969378; 25. When the given number is any COMMON FRACTION. Reduce the given fraction to a decimal, and find its logarithm, as in Art. 24; or, since a fraction is an expression of division, subtract the logarithm of the denominator from the logarithm of the numerator, and the difference will be the logarithm of the fraction (Art. 10). Or, Thus, the logarithm of, or .75, is 1.875061. the logarithm of the numerator, 3, is 0.477121 TO FIND THE NUMBER CORRESPONDING TO ANY LOGARITHM. 26. When the given logarithm is WITHIN the limits of the table. Look in the column headed 0, for the first two or three figures of the logarithm, neglecting the characteristic, and, if all the figures of the logarithm are found in that column, the corresponding number will be on the same horizontal line, in the column headed N. If, however, the decimal part of the logarithm be not exactly found in the column headed 0, look for it in one of the nine following columns, and the first three figures of the corresponding number will be on the same horizontal line in the column headed N, and the fourth will be at the head of the column in which the logarithm was found. Make the number correspond with the characteristic given, if necessary, by pointing off decimals or annexing ciphers (Arts. 7 and 8). Thus, the number corresponding to the logarithm 3.146128 is 1400; 0.370143 "2.345; 66 66 27. When the given logarithm is NOT WITHIN the limits of the table. From the decimal part of the given logarithm subtract the decimal part of that next less; annex to their difference two or more ciphers, and divide by the number, in the column headed D, opposite the decimal part taken from the table. Annex the result to the number corresponding to the lesser logarithm, and point off according to the characteristic, as before. It sometimes happens, in dividing by the tabular difference, that there are not as many figures in the quotient as there are ciphers annexed to the dividend. In such a case, supply the deficiency, as in the division of decimals, by prefixing a cipher or ciphers to the quotient before annexing. This process, like its converse (Art. 23), is based upon the supposition that the differences of logarithms are proportional to the differences of their corresponding numbers. NOTE. The number corresponding to a given logarithm, when obtained by the use of a table calculated to six decimal places, is reliable only to the sixth figure, and sometimes that figure of an answer is not strictly correct. Let it be required to find the number corresponding to the logarithm 2.633356. The dec. part of the given log. is .633356 log. next less is .633266, correspon. num., 4298 28. The arithmetical complement of any logarithm is the difference between it and 10. Thus, the arithmetical complement of the logarithm of 41, is 10-log 41 = 10 — 1.612784 = 8.387216. 29. The arithmetical complement of a logarithm may be readily found, from the table, by subtracting each figure of the logarithm from 9, excepting the last significant figure at the right, which must be taken from 10; for this is equivalent to subtracting the logarithm from 10. 30. The difference of two logarithms is equal to the sum of the logarithm to be diminished and the arithmetical complement of the other, less 10. For let a be any logarithm, and b a logarithm to be subtracted from it; then their difference will be a- b. Now the arithmetical complement of b is 10-b; adding 10-b to a, we have a + 10-b; subtracting 10, we have |