Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

EXAMPLE.-Solve this expression by logarithms:
497 x .0181 X 762

?
3,300 X .6517
SOLUTION.- Log 497 2.69636

.0181

2.25768 762 = 2.88195

Log
Log

Log product = 3.83599
Log 3,300 3.51851
Log .6517 7.81405

Log product 3.33256
3.83599 - 3.33256 .50343 log 3.1874.

497 x .0181 X 762 Hence,

= 3.1874.
3,300 X .6517
EXAMPLE.-Solve

504,203 X 507
V1.75 X 71.4 X 87

by logarithms. SOLUTION.- Log 504,203 = 5.70260

Log 507 = 2.70501

3

[blocks in formation]

3

Log product 4.03626 8.40761 -- 4.03626

1.45712 log 28.65.

3 Hence, 504,203 X 507

= 28.65. 1.75 X 71.4 x 87 Logarithms can often be applied to the solution of equations.

EXAMPLE.-Solve the equation 2.43.ch 0.0648.
SOLUTION.-

2.43.5 .0648.

ů.0648 Dividing by 2.43,

2.43 Taking the logarithm of both numbers,

log .0648 5 X log x =

log 2.43;

6

or

X =

2.81158
5 log x =

.38561
6
1.80193 .38561

1.41632. Dividing by 5, log x 1.88326; whence,

20 = .7643.
EXAMPLE.-Solve the equation 4.5% 8.
SOLUTION.—Taking the logarithms of both numbers,

x log 4.5 = log 8,
log 8

.90309 whence,

log 4.5

.65321
Taking logarithms again,
log x = log .90309 – log .65321 1.95573 – 1.81505

.14068; whence, it = 1.3825. REMARK.-Logarithms are particularly useful in those cases when the unknown quantity is an exponent, as in the last example, or when the exponent contains a decimal, as in several instances in the examples given on pages 45-49. Such examples can be solved without the use of logarithms, but the process is very long and somewhat involved, and the arithmetical work required is enormous. To solve the example last given without using the logarithmic table and obtain the value of x correct to five figures would require, perhaps, 100 times as many figures as were used in the solution given, and the resulting liability to error would be correspondingly increased; indeed, to confine the work to this number of figures would also require a good knowledge of short-cut methods in multiplication and division, and judgment and skill on the part of the calculator that can only be acquired by practice and experience.

Formulas containing quantities affected with decimal exponents are generally of an empirical nature; that is, the constants or exponents or both are given such values as will make the results obtained by the formulas agree with those obtained by experiment. Such formulas occur frequently in works treating on thermodynamics, strength of materials, machine design, etc.

COMMON LOGARITHMS

6 7 8

112

N. L. 0 1

4

9

P. P. 100 100 000 043 087 130 173 217 260 303 346 389 101 432 475 518 561 604 647 689 732 775 817

44 | 43

42 102 860 903 945) 988 *030 *072*115 *157 *199 *242

1 4.4 4.3 4.2 103 01 284 326 368 410 452 494 536 578 620 662

2 8.8 8.6 8.4 104 703 745 787 828 870 912 953 995 036 #078

13.2 12.9 12.6 105 02 119 160 202 243 284 325 366 407 449 490 4 17.6 | 17.2 16.8 106 531 572 612 653 694 735 776 816 857 898

22.0 21.5 21.0 107 938 979 *019 *060 *100*141*181 *222 +262 *302 6 26.4 25.825.2 108 03 342 383 423 463 503 543 583 623 663 703

7 30.8 30.1 29.4 743 782 822 862 902 941 981 *021 *060 *100

35.2 34.4 33.6

9 39.6 38.7 | 37.8 10 04 139 179 218 258 297 336 376 415 454 493 111 532 571 610 650 689 727 -766 805 844 883

41 | 40 39 9221 961 999 #038 *077 *115 *154 *192 *231 +269 11 4.1 4.0 3.9 113 05 308 346 385 423 461 500 538 576 614 652

2
8.2

8.0 7.8 114 690 729 767) 805 843 881 918 956 994 *032

3 12.3 12.0 11.7 115 06 070 108 145 183 221] 258) 296 333 371 408

16.4 16.0 15.6 116 446 483 521 558 595 633 670 707 744 781

5 20.5 20.0 19.5 117

819 856 893 930 967 *004 *041 *078 *115 *1511 6 24.6 24.0 23.4 118007 188 225 262 298 335 372 408 445 482 5181 7 28.7 28.0 27.3 119 555 591 628 664 700 737 773 809 846 882

8 32.8 32.0 31.2

9 36.936.0 35.1 120 918 954 990 *027 *063 *099 *135 *171 *207 *243 121|08 279 314 350 386 422) 458, 493 529 565 600

38 37 36 122 636 672 707 743 778 814 849 884 920 955

1 3.8 3.7 3.6 123 991 *026 *061 *096 *132*167 *202 *237 *272 *307

2
7.6

7.4 7.2 124 09 342 377 412 447 482 517 552 587 621 656

3 11.4 11.1 10.8 125

691 726 760 795 830 864 899 934 968 *003 4 15.2 | 14.8 14.4 126 10 037 072 106 140 175) 2091 2431 2781 312 346 5 19.0 18.5 18.0 127 380 415 449 483 517| 551 585 619 653 687 6 22.8 22.221.6 128 721 755 789 823 857 890 924 958 992 *025

7 26.6 25.925.2 129 11 059 093 126 160 193 2271 261 294 327 361 8 30.4 29.6 28.8

9 34.2 33.3 I 32.4 130

394 428 461 494 528 561 594 628 661 694 131 727 760 793 826 860 893 926 959 992 *024

35 34

33 132 12 057 090 123 156 189 222 254 287 320 352 1 3.5

3.3 133 385 418 4501 483 516) 548 581 613 646 678

2 7.0 6.8 6.6 134 710 743 775 808 840 872 905 937 969 *001 10.5 10.2 9.9 135/13 0331 066 098 1301 162 194 226 258 290 322

4 14.0 13.6 13.2 136 354 386 418 450 481 513 545 577 609 640

5 17.5 17.0 16.5 137 672 704 735 767 799 830 862 893 925 956 6 21.0 20.4 19.8 138 988 *019 *051 *082 *114*145 *176 *208 *239 *270

7 24.5 23.8 23.1 139 14 301 333 364 395 426 457 489 520 551 582

8 28.0 | 27.2 26.4

9 31.5 30.6 29.7 140

613 644 675 706 737 768 799 829 860 891 141 922 953 983 *014 *045 *076 *106 *137 *168 *198

32 31 30 142 15 229 259 290 320 351 381 412 442 473 503

1 3.2 3.1 3.0 143 534 564 594 625 655 685 715 746 776 806

2 6.4 6.2 6.0 144 836 866 897 927 957| 987 *017 *047 *077 *107

9.6 9.3 9.0 145 16 1371 167 197) 227 256 286 316 346 376 406

4 12.8 12.4 12.0 146 435 465) 495 524 554 584 613 643 673 702

5 16.0 15.5 15.0 147 732 761 791 820 850 879 909 938 967 997

6 19.2 18.6 18.0 148 17 026 056 085 114 143 173 202 231 260 289

7 22.4 | 21.7 21.0 149 319 348 377 406 435 464 493 522 551 580

8 25.6 24.8 24.0 150 609 638 667 696 725 754 782 811 840 869

9 28.8 | 27.9 | 27.0 N. L. O 1 4 5 6 7 8 9

P. P.

3.4

2 3

[blocks in formation]
[blocks in formation]

162

1 2 3 4 5 6 7 8 9

5.4 5.2 8.1 7.8 10.8 10.4 13.5 13.0 16.2 15.6 18.9 18.2 21.6 20.8 24.3 23.4

25

2.5

150 17 609 638 667 696 725 754 782 811 840 869 151

898 926 955 984 *013 *041 *070 *099 *127 *156 152 18 184 213 241 270 298 327 355 384 412 441 153 469 498 526 554 583 611 639 667 696 724 154 752 780 808 837 865 893 921 949 977 *005 155 19 033 061 089 117 145173 201 229 257 285 156 312 340 368 396 424 451 479 507 535 562 157 590 618 645 673 700 728 756 783 811 838 158

866 893 921 948 976 *003 *030 *058 *085 112 159 20 140 1671 194 222 249 276 303 330, 358 385 160 412 439 466 493 520 548 575 602 629 656 161 683 710 737 763 790 817 844 871 898 925

952 978 *005 *032 *059 *085 *112 *139 *165 *192 163/21 219 245 272 299 325 352 378 405 431 458 164 484 511 537 564 590 617 643 669 696 722 165 748 775 801 827 854 880 906 932 958 985 166/22 011 037) 063 089 115| 141 167 194 220 246 167 272 298) 324! 350 376 401) 427 453 479 505 168 531 557 583 608 634 660 686 712 737 763

169 789 814 840 866 891 917 943 968 994 *019 170 23 045 070 096 121 147) 172 198 223 249 274 171 800 325 350 376 401 426 452 477 502 528 172

553 578 603 629 654 679 704 729 754 779 173 805 830 855 880 905 930 955 980 *005 *030 174 24 055 080 105 130 155 180 204 229 254 279 175 304 329 353 378 403 428 452 477 502 527 176 551 576 601 625 650 674 699 724 748 773 177 797| 822 846 871 895| 920 944 969 993 *018 178 25 042 066 091 115 139 164 188 212 237 261

179 285) 310 834 358 382 406 431 455 479 503 180 527 551 575 600 624 648 672 696 720 744 181

768 792 816 840 864 888 912 935 959 983 182 26 007 031 055 079 102 126 150 174 198 221 183 245 269 293 316 340 364 387 411 435 458 184 482 505 529 553) 576 600 623 647 670 694 185 717 741 764 788 811 834 858 881 905 928 186

951 975 998 *021 *045 068 *091 *114 *138 *161 187 27 184 207 231 254 277) 300 323 346 370 393 188 416) 439| 462 485 508 531 554 577 600 623 189 646 669 692 715 738/ 761 784 807 830 852 190 875 898 921 944 967 989 *012 *035 *058'*081

191 28 103 126 149 171 194217 240 262 285 307 192 330 353 375 398 421 443 466 488 511 533 193

556 578 601 623 646 668 691 713 735 758 194 780 803) 825 847 870 892 914 937 959 981 195 29 003 026 048 070 092) 115 137 159 181 203 196 226 248 270 292 314 336 358 380 403 425 197 447) 469 491 513 535 557 579 601 623 645 198 667 688 710 732 754 776 798 820 842 863 199

1 2 4 4 5 6 7 8 9

5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5

[blocks in formation]

1
2
3
4
5

6.6

885907 929 951 973 994 *016 #038 *060 *081 200 30 103 125 146 168 190 211 233 255 276 298 N. LO 1 2 3 4 5 6 7 8 9

6.3 8.8 8.4 11.0 10.5 13.2 12.6 16.4 | 14.7 17.6 16.8 19.8 18.9

7 8 9

P.P.

TABLE-( Continued).

P. P.

202

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

N. L. 0 | 1 2 3 4 56 7 8 9 200 30 103 125 146 168 190 211 233 255 276 298 201 320 341 363 384 406 428 449 471 492 514

535 557 578) 600 621 643 664 685 707 728 203 750) 771 792 814 835 856 878 899 920 942 204 963 984 006 *027 *048 *069 *091 *112 *133 154 205 31 175 197 218 239 260 281 302 323 345 366 206 387 408 429 450 471 492 513 534 555 576 207

597 618 639 660 681 702 723 744 765 785 208 806 827 848 869 890 911 931 952 973 994

209 32 015 035 056 077 098| 118 139 160 181 201 210

222 243 263 284 305 325 346 366 387 408 211 428 449 469 490 510 531 552 572 593 613 212 634 654 675 695 715 736 756 777 797 818 213

838 858 879 899 919 940 960 980 *001 *021 21433 041 062 082 102 122 143 163 183 203 224 215 244 264 284 304 325 345 365 385 405 425 216 445 465 486 506 5261 546 566 586 606 626 217 646 666 686 706 726 746 766 786 806 826 218 846 866 885 905 925945 965 985 *005 *025

219|34 044 064 084 104 124 143 163 183 203 223 220 242 262 282 301 321) 341 361 380 400 420

221 439 459 479 498 518 537 557 577 596 616 222 635 655 674 694 713 733 753 772 792 811 223 830 850 869 889 908 928 947 967 986 *005 22435 025 044 064 083 102] 122 141 160 180 199 225 218 238 257 276 295 315 334) 353 372 392 226 411 430 449 468 488 507 526) 545 564 583 227 603 622 641 660 679 698 717 736 755 774 228 793 813 832 851 870) 889 908 927 946 965

229 984'*003 *021 *040 *059 *078 *097 *116 *135 *154 230 36 173 192 211 229 248 267 286 305 324 342

231
232 549 568 686 605 624 642 661 680 698 717
233 736 754 773 791 8101 829 847 866 884 903
234 922 940 959 977 996 *014 *033 *051 *070 *088
235 37 1071 125 144 162 181 199 218 236 254 273
236 291 310 328 346 365 383 401 420 438 457
237 475 493 511 530 5481 566 585 603 621 639
238 658, 676 694 712 731) 749 767 785 803) 822

239 840 858 876 894 912 931 949 967 985 *003 240 38 021 039 057 075 093 112 130 148 166 184

241 202 220 238 256 274 292 310 328 346 364 242 382 399 417 435 453 471 489 507 525 543 243 561 578 596 614 632 650 668 686 703 721

739 757 775 792 810 828 846 863 881 899 245 917 934 952 970 987 *005 *023 *041 *058 *076 246|39 094 111 129 146 164 182 199 2171 235 252 247 270 287 305 322 340 358 375 393 410 428 248 445 463 480 498 515) 533 550 568 585 602 249 620 637 655 672 690 707 724 742 759 777 250 794 811 829) 846 863 881 898 915 933 950 N. L.0 1 2 3 4 5 6 7 8 9

22 21 2.2 2.1 4.4 4.2 6.6 6.3 8.8 8.4 11.0 10.5 13.2 12.6 15.4 | 14.7 17.6 16.8 19.818.9

20 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 19 1.9 3.8 5.7 7.6 9.6 11.4 13.3 15.2 17.1

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

18 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2

244

[blocks in formation]
« ΠροηγούμενηΣυνέχεια »