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N

10

11

12

13

14

15

16

01 2 3 4 5 6 7 89

0000 0043 0086 0128 0170 0212 0253 0294 0334 0374

0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732

1761 1790 1818 1847 1875 1903 1931 1959 1987 2014

2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 | 2648 | 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 | 2878 2900 2923 2945 2967 2989

20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201

21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962

25

26

3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 4150 4166 4183 4200 4216 4232 4249 | 4265 | 4281 4298 27 4314 4330 4346 4362 | 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757

30

4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 | 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 | 5428

35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010

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40

6021 6031 6042 6053 6064 6075 6085 6096 6107 6117

41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522

45

46

6532 6542 6551 6561 6571 6580 6590 6599 6609 6618

6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981

49

50

6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396

55

7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122

65

8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 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 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025

75

76

177

78

79

80

81

9031 9036 9042 9047 9053 9058 9063 9069 9074 9079

9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 9243 9248 9253 9258 9263 9269 0274 9279 9284 9289

84

9294 9299 9304 9309 9315 9320 9325 9330 9335 9340

85 92

86

9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538

90

9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 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 56.7=1.7536.

(b) 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.

[blocks in formation]

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

[blocks in formation]

(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 of the way from 62500 to 62600; hence if to the logarithm of 62500 we add of 0007, a nearly correct logarithm of 62543 is obtained.

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=4·7202

log 0005240=4·7193

differences ⚫0000010 ⚫0009

Now 0005243 is 0000003 greater than 0005240; hence log 0005243 equals log·0005240 plus (the difference of logarithms);

⚫0000003 3
or of ⚫0009
•0000010 10

that is,

log 0005243=4·7193

•0003 (nearly)

-4.7196

In practice negative characteristics are usually avoided by adding them to 10 and writing -10 after the logarithm. Thus in the above example 4·7196=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 tablēs 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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