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N

0 1 2

2 3 4 5 6 7

8 9

10

0000 0043 0086 0128 0170 0212 0253 0294 0334 0374

11 12 13 14

04140453 049205310569 0607 064506820719 0755 07920828 0864 0899 0934 0969 1004 | 1038 1072 1106 1139 1173 1206 1239 1271 1303 | 1335 | 1367 | 1399 1430 1461 1492 1523 15531584 1614 1644 1673 1703 1732

15

1761 1790 1818 1847 1875 | 1903 1931 | 1959 1987 2014

16 17 18 19

2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989

20 || 3010 3032 3054 3075 3096 3118 | 31393160 | 3181 3201

21 22 23 24

32223243 3263 3284 3304 | 3324 3345 3365 3385 3404 34243444 3464 3483 3502 3522 3541 3560 3579 3598 3617 3636 3655 3674 | 3692 | 3711 3729 | 3747 3766 3784 3802 3820 3838 | 3856 3874 3892 3909 3927 3945 3962

25

3979 3997 | 4014 4031 4048 4065 4082 | 4099 4116 4133

26 27 28 29

4150 4166 | 4183 4200 4216 4232 42494265 4281 | 4298 4314 4330 4346 4362 4378 4393 4409 | 4425 4440 4456 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757

30 || 4771 4786 4800 4814 4829 4843 | 4857 | 4871 4886 4900

31 32 33 34

4914 4928 4942 4955 4969 4983 4997 | 5011 5024 | 5038 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 5185 5198 | 5211 5224 5237 5250 5263 5276 5289 | 5302 5315 5328 5340 5353 | 5366 5378 5391 5403 5416 5428

35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551

36 37 38 39

5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 5798 5809 5821 58325843 5855 5866 5877 5888 5899 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010

N

0 | 1 | 2 3

4 5

6 7 8 9

40

60216031 | 6042 60536064 6075 | 6085 6096 6107 6117

41 42 43 44

6128 6138 6149 6160 | 6170 6180 61916201 62126222 62326243 6253 62636274 6284 6294 6304 6314 6325 6335 6345 6355 6365 | 6375 6385 6395 | 6405 | 6415 6425 6435 6444 6454 6464 6474 | 6484 64936503 6513 6522

45

6532 6542 6551 6561 6571 6580 6590|6599 6609 6618

46 47 48 49

6628 6637 6646 66566665 6675 6684 6693 6702 6712 672167306739 6749 | 6758 6767 | 6776 6785 67946803 6812 6821 6830 | 6839 6848 6857 | 6866 6875 6884 6893 6902 6911 6920 6928 6937 6946 6955 6964 | 6972 6981

50 6990 1998 7007 7016 7024 7033 7042 7050 7059 7067

51 52 53 54

7076 7084 7093 7101 7110 71187126 7135 7143 7152 7160 7168 7177 | 7185 7193 7202 7210 7218 7226 7235 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 7324 7332 7340 73487356 7364 7372 | 7380 7388 7396

55

7404 7412 7419 7427 7435 7443 7451 7459 7466 7474

56 57 58 59

7482 7490 74977505 7513 7520 7528 7536 7543 7551 7559 7566 7574 | 7582 7589 7597 | 7604 76127619 7627 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 7709 7716 7723 7731 7738 7745 7752 7760 | 7767 7774

60 | 7782 77897796 7803 7810 7818 7825 7832 78397846

61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 | 7938 7945 7952 7959 79667973 7980 7987 63 7993 8000 8007 | 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 80828089 | 8096 81028109 8116 8122

65

81298136 8142 8149 8156 | 8162 81698176 8182 8189

66 || 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 82808287 8293 82998306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 83888395 8401 84078414 8420 8426 8432 8439 8445

N

0 1 2 3

4 5 6 7 8 9

70 || 8451 8457 8463 | 8470 8476 8482 8488 8494 8500 8506

71 72 73 74

85138519 8525 8531 8537 8543 8549 8555 8561 8567 8573 8579 | 8585 8591 8597 8603 86098615 8621 8627 8633 8639 8645 | 8651 | 8657 8663 8669 8675 8681 8686 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745

75 || 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802

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76 || 8808 8814 8820 8825 8831 8837 | 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 | 8893 8899 8904 8910 8915 78 | 8921 8927 8932 8938 | 8943 8949 8954 8960 8965 8971 79 || 8976 | 8982 8987 | 8993 8998 9004 90099015 9020 9025

80 || 9031 9036 9042 9047 9053 9058 9063 9069 | 9074 9079

81 9085 90909096 9101 9106 9112 91179122 9128 9133 82

91389143 9149 9154 9159 9165 9170 | 9175 9180 9186 83 91919196 9201 9206 9212 9217 | 9222 9227 9232 9238 84 | 9243 9248 9253 9258 9263 9269 0274 9279 9284 9289

85 92

85 9294 | 9299 9304 9309 9315 9320 9325 9330 9335 9340

86 || 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 | 9400 9405 9410 | 9415 9420 9425 94309435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 || 9494 9499 9504 9509 | 9513 9518 9523 9528 9533 9538

90 || 95429547 9552 9557 9562 9566 9571 9576 9581 9586

91 92 93 94

95909595 9600 9605 9609 9614 96199624 9628 9633 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 9685 96899694 9699 9703 9708 97139717 9722 9727 9731 9736 9741 | 9745 9750 9754 9759 9763 9768 9773

95

9777 | 9782 9786 9791 9795 9800 9805 9809 9814 9818

96 97 98 99

9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 9868 9872 9877 9881 9886 9890 9894 98999903 9908 9912 9917 9921 9926 9930 9934 99399943 | 9948 9952 9956 9961 9965 99699974 9978 9983 9987 9991 9996

USE OF THE TABLE.

416. On pages 334–336 we give a four-place table containing the mantissæ of the common logarithms of all integers from 100 to 1000.

To FIND THE LOGARITHM OF A NUMBER. (a) Suppose the number consists of three figures, as 56-7.

In the column headed N find the first two significant figures. On a line with these and in the column having at the top the third figure will be found the mantissa. Thus on a line with 56 and in the column headed 7 we find 7536. To this, which is the decimal part of the logarithm, prefix the characteristic [Art. 408], and we have

log 567=1.7536. (6) Since in common logarithms the mantissa remains unchanged when the number is multiplied by an integral power of 10, we change one or two-figure numbers into three-figure numbers by addition of ciphers before looking for the mantissæ. The mantissa of log 56 will be that of 560, the only change in the logarithm being in the characteristic. Thus

log 560=2.7482,

log 56=1.7482. In the same manner log 7 has for mantissa that of log 700.

log 700=2.8451,

log 7=0·8451. (c) Suppose the logarithm of a number of more than three figures, as 62543, is required. Since the number lies between 62500 and 62600, its logarithm lies between their logarithms. In the column headed N we find the first two figures, 62; on a line with these and in the columns headed 5, and 6, we find the mantissæ 7959 and ·7966. Prefixing the characteristic [Art. 408], we have

log 62600=4.7966,

log 62500=4.7959. Therefore while the number increases from 62500 to 62600, the logarithm increases •0007. Now our number is 10% of the way from 62500 to 62600; hence if to the logarithm of 62500 we add 10% of .0007, a nearly correct logarithm of 62543 is obtained.

or

Thus

log 62543=4.7959

.0003 correction

=4.7962 (d) Suppose the logarithm of a decimal, as •0005243, is required. The number lies between .0005240 and ·0005250. In the column headed N we find the first two significant figures, 52; on a line with these and in the columns headed 4, and 5, we find the mantissæ :7193 and 7202. Prefixing the characteristic [Art 409], we have

log •0005250=2.7202

log •0005240=2.7193

differences .0000010 ·0009 Now •0005243 is :0000003 greater than •0005240; hence

.0000003 3 log-0005243 equals log .0005240 plus

of .0009

.0000010 10 (the difference of logarithms); that is,

log •0005243=2.7193

·0003 (nearly)

=7.7196 In practice negative characteristics are usually avoided by adding them to 10 and writing – 10 after the logarithm. Thus in the above example 2.7196 =6.7196 – 10.

417. The increase in the logarithms on the same line, as we pass from column to column, is called the tabular difference. In finding the logarithm of 62543, we assumed that the differences of logarithms are proportional to the differences of their corresponding numbers, which gives us results that are approximately correct. For greater accuracy we must use tables of more places.

To FIND THE NUMBER CORRESPONDING TO A LOGARITHM.

418. (a) Suppose a logarithm, as 1.7466, is given to find the corresponding number.

Look in the table for the mantissa •7466. It is found in the column headed 8 and on the line with 55 in the column headed N. Therefore we take the figures 558, and, as the characteristic is 1, point off two places, obtaining the number 55.8.

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