397. For the same sequence of digits the mantissa does not ... log.003964 = log 3.964 + log .001 = .59813+(-3)= 7.59813 – 10, and, in general, if a is any positive or negative integer, log (10a × m)= a x log 10+ log m = a + log m. 398. As any number multiplied by a power of 10 will differ only in position of the decimal, the log of such number will always be the log of the power of 10 plus the log of the number, but the log of the power of 10 is always integral, hence the decimal part of the result will not be changed. 399. This property, which belongs only to the common system, makes it the only system suited to numerical calculation, and gives it great superiority over other systems; it may be stated as follows: S 400. I. The characteristic depends on the position of the decimal point. II. The mantissa depends on the sequence of digits. 401. If the logarithm of a number to one base is given, find the logarithm of the same number to another base. Let a and b be the bases of two systems, and m the number whose log to base a is known, then we have, Find the value of the following to the base of 10: 1. log212. 2. log, 14. 3. log, 29. 5. Find the logarithm of 324 to the base of 5. 6. Find the logarithm of to the base of 9. 4. log. 51. EXPLANATION OF TABLES. 402. In the tables given on pp. 274-275, the mantissa of the logarithms of all numbers between 1 and 1000 are given to four places of decimals. 403. To find the log of a number of one or two significant places, use the column headed 0. Thus, log 59 = 1.7709, the mantissa of 59 and 590 being identical. The characteristic is 1. Again, log 9 = .9542, being the same as the mantissa of 900. NOTE. The first figure of the mantissa in the table is not repeated, except in the horizontal line in which it first occurs. 404. To find the logarithm of a number of three significant places. In the column headed No. find the first two significant places. In the horizontal line with these two significant figures, and in the column headed with the third significant figure, will be found the mantissa of the logarithm required. Prefix the proper characteristic. Thus, the log 864 = 2.9365, as .9365 occurs in the horizontal line of 86 and in column headed 4. As the number has three places to the left of the decimal point, the characteristic is 2. 405. To find a logarithm of a number of more than three figures. Find the logarithm 36595. In line 36, column 5, is found .5623, and in line 36, column 6, is found .5635. The difference between these mantissas is .0012. The difference between 36500 and 36600 is 100, and the difference between 36500 and 36595 is 95. Therefore 190 of .0012, or 0011, must be added to .5623 to produce the required mantissa, which is .5634. 95 414 453 492 792 828 864 1139 1173 1206 14 461 492 523 0170 0212 531 569 607 645 682 719 755 899 934 969 1004 1038 1072 1106 1239 1271 1303 335 367 399 430 553 584 614 644 673 703 0253 0294 0334 0374 732 15 1761 1790 1818 1847 16 2041 2068 2095 2122 304 330 355 380 553 577 601 625 788 810 833 856 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 693 702 712 776 785 794 803 964 866 875 884 893 972 981 50 6990 6998 7007 7016 7024 51 7076 7084 093 101 110 52 160 168 177 185 193 7033 7042 7050 118 126 135 202 210 218 275 284 292 300 364 372 380 |