Suppose, however, that the given number has more than five digits. For example: Suppose we require to find log 62761.6. We find from the table log 62761 = 4.7976899 log 62762 = 4.7976968 diff. for 1 = 0.0000069 and Thus for an increase of 1 in the number there is an increase of .0000069 in the logarithm. Hence, assuming that the increase of the logarithm is proportional to the increase of the number, then an increase in the number of .6 will correspond to an increase in the logarithm of .6 × .0000069.0000041, to the nearest seventh decimal place. This explains the use of the column of proportional parts on the extreme right of the page. It will be seen that the difference between the logarithms of two consecutive numbers is not always the same; for instance, those in the upper part of the page before us differ by .0000070, while those in the middle and the lower parts differ by .0000069 and .0000068. Under the column with the heading 69 we see the difference 41 corresponding to the figure 6, which implies that when the difference between the logarithms of two consecutive members is .0000069, the increase in the logarithm corresponding to an increase of .6 in the number is .0000041; for .06 it is evidently .0000004, and so on. NOTE. We assume in this method that the increase in a logarithm is proportional to the increase in the number. Although this is not strictly true, yet it is in most cases sufficiently exact for practical purposes. Had we taken a whole number or a decimal, the process would have been the same. N. 6250 51 P.P. 1 2 3 4 5 6- 7. 8 9 795 8800 8870 8939 9009 9078 9148 92179287,9356-9426 9495 9564 9634 9703 9773 9842 9912 9981 0051 0120 796 0190 0259 0329 0398 0468 0537 0606 0676 0745 0815 0884 0954 1023 1093 1162 1232 1301 1370 1440 1509| 1579 1648 1718 1787 1857 1926 1995 2065 2134 2204 2273 2343 2412 2481 2551 2620 2690 2759 2829 2898 2967 3037 3106 3176 3245 3314 3384 3453 3523 3592 3662 3731 3800 3870 3939 4009 40784147 4217 4286 1 4356 4425 4494 4564 4633 4703 4772 4841 4911 4980 2 14.0 5050 5119 5188 5258 5327 5396 5466 5535 5605 5674 3 21.0 6260 796 5743 5813 5882 5951 6021 6090 6160 6229 6298 6368 4 28.0 52 53 54 55 56 57 58 59 61 62 63 64 65 66 70 7.0 6437 6506 6576 6645 6714 6784 6853 6923 6992 7061 5 35.0 7131 7200 7269 7339 7408 7477 7547 7616 7685 7755 6 42.0 7824 7893 7963 8032 8101 8171 8240 8309 8379 8448 7 49.0 8517 8587 8656 8725 8795 8864 8933 9003 9072 9141 8 56.0 9211 9280 9349 9419 9488 9557 9627 9696 9765 9835 9 63.0 9904 9973 0043 0112 0181 0250 0320 0389 0458 0528 67 797 0597 0666 0736 0805 0874 0943 1013 1082 1151 1221 1290 1359 1428 1498 1567 1 636 1706 1775 1844 1913 1983 2052 2121 2191 2260 2329 2398 2468 2537 2606 6270 797 2675 2745 2814 2883 2952 3022 3091 3160 3229 3299 3368 3437 3507 3576 3645 3714 3784 3853 3922 3991 6137 6207 6276 6345 6414 6483 6553 6622 6691 6760 4 27.6 6829 6899 6968 7037 7106 7175 7245 7314 7383 7452 5 34.5 7521 7590 7660 7729 7798 7867 7936 8006 8075 8144 6 41.4 8213 8282 8351 8421 8490 8559 8628 8697 8766 8836 7 48.3 8905 8974 9043 9112 9181 9251 9320 9389 9458 9527 8 55.2 6280 797 9596 9666 9735 9804 9873 9942 0011 0080 0150 0219 9 62.1 81 798 0288 0357 0426 0495 0565 0634 0703 0772 0841 0910 82 83 84 85 86 87 88 89 0979 1048 1118 1187 1256 1325 1394 1463 1532 1601 68 3053 3122 3191 3260 3329 3398 3467 3536 3606 3675 3744 3813 3882 3951 4020 4089 4158 4227 4296 4366| 4435 4504 4573 4642 47114 780 4849 4918 4987 5056 1 6.8 5125 5194 5263 5333 5402 5471 5540 5609 5678 5747 2 13.6 5816 5885 5954 6023 6092 6161 6230 6299 6368 6437 3 20.4 6290 798 6506 6575 6645 6714 6783 6852 6921 6990 7059 7128 4 27.2 7197 7266 7335 7404 7473 7542 7611 7680 7749 7818 5 34.0 7887 7956 8025 8094 8163 8232 8301 8370 8439 8508 6 40.8 8577 8646 8715 8784 8853 8922 8991 9060 91299198 7 47.6 9267 9336 9405 9474 9543 9612 9681 9750 9819 9888 8 54.4 9957 0026 0095 0164 0233 0302 0371 0440 0509 0578 9 61.2 799 0647 0716 0785 0854 0923 0992 1061 1130 1199 1268 91 92 93 94 95 96 97 98 99 1337 1406 1475 1544 1613 1682 1751 1820 1889 1958 2027 2096 2164 2233 2302 2371 2440 2509 2578 2647 2716 2785 2854 2923 2992 3061 3130 3199 3268 3337 6300 799 3405 3474 3543 3612 3681 3750 3819 3888 3957 4026 2 3 4 5 6 7 8 9 P.P. N. Thus, suppose we require to find log 627616 and log .627616. The mantissa is exactly the same as before (Art. 66), and the only difference to be made in the final result is to change the characteristic according to rule (Art. 64). 69. To find the Number corresponding to a Given Logarithm. If the decimal part of the logarithm is found exactly in the table, we can take out the corresponding number, and put the decimal point in the number, in the place indicated by the characteristic. Thus if we have to find the number whose logarithm is 2.7982915, we look in the table for the mantissa .7982915, and we find it set down opposite the number 62848: and as the characteristic is 2, there must be one cipher before the first significant figure (Art. 64). Hence 2.7982915 is the logarithm of .062848. Next, suppose that the decimal part of the logarithm is not found exactly in the table. For example, suppose we have to find the number whose logarithm is 2.7974453. We find from the table Thus for a difference of 1 in the numbers there is a difference of .0000069 in the logarithms. The excess of the given mantissa above .7974407 is (.7974453-.7974407) or .0000046. Hence, assuming that the increase of the number is proportional to the increase of the logarithm, we have .0000069: .0000046:: 1: number to be added to 627.25. We might have saved the labor of dividing 46 by 69, by using the table of proportional parts as follows: 69 a. Arithmetic Complement. By the arithmetic complement of the logarithm of a number, or, briefly, the cologarithm of the number, is meant the remainder found by subtracting the logarithm from 10. To subtract one logarithm from another is the same as to add the cologarithm and then subtract 10 from the result. Thus, a-ba (10b) - 10, where a and b are logarithms, and 10-b is the arithmetic. complement of b. When one logarithm is to be subtracted from the sum of several others, it is more convenient to add its cologarithm to the sum, and reject 10. The advantage of using the cologarithm is that it enables us to exhibit the work in a more compact form. The cologarithm is easily taken from the table mentally by subtracting the last significant figure on the right from 10, and all the others from 9. |