value. With such a base the logarithms of numbers greater than 1 will be negative, less than 1 positive, Thus, with as the base, the log ofis 2; of is 3 "81" -3 264. The logarithms of numbers which form a geometrical series form an arithmetical series. For, if a series increased or decreased by a constant ratio, its logarithms would increase or decrease by a constant difference equal to the logarithm of the constant ratio. For an example see Art. 243; here the numbers decrease by the constant ratio 10, and the logarithms by the constant difference 1. 265. From the principles of the previous articles it will be easy to find the logarithms of the perfect roots and powers of any number. Thus, 1. In a system whose base is 8, 2. In a system whose base is 4, what is the logarithm of 4 of 16 of 64 of 2 of 8 of 1? of? of? of? of 0 ? 3. In a system whose base is 9, what is the logarithm of 81 of 3 of 27 of 9 of 1? of? of? of 0? 4. In a system whose base is, what is the logarithm of 2? of 32 of 8 of ? of? 5. If the logarithm of 0.125 is 2.5, what is the base? x = 0.125 * = 0.125 ̄ = (}; ) * = 83 = 4, Ans. = 6. If the logarithm of 0.5 is 1.8, what is the base? Ans. 32. 7. If the logarithm of 0.3 is 0.3, what is the base? J EXPONENTIAL EQUATIONS. 266. An equation having the unknown quantity as an exponent, or an exponential equation, may be solved by means of logarithms. 268 LOGARITHMS OF NUMBERS. NO1 2 3 4 5 6 7 8 9 PROPORTIONAL PARTS. 2 3 4 5 6 7 8 9 8/3 12.4 16.6 20.7 24.8 29.0 33.1 37-3 3.5 10 0000 043 086 128 170 212 253 294 334 374 4.1 20 3010 032 054 075 096 118 139 160 181 201 3.0 8.4 11.2 14.0 16.8 19.6 22.4 25.2 2.5 5.0 7-49.9 2.4 14-9 17.4 19.9 22.3 1.9 3.9 5.8 8.5 10.6 12.7 14.8 17.0 19.1 8.1 10.1 12.1 14.1 16.2 18.2 7.7 1.8 3.7 5.5 7.4 1.8 3.5 5.3 7.1 1.7 3.4 5.1 6.8 1.6 3.3 4.9 6.6 1.6 3.2 4.7 6.3 1.5 3.0 4.6 6.1 1.5 2.9 4.4 5.9 7.4 9.7 11.6 13.5 15.4 17.4 9.2 11.1 12.9 14.8 16.6 8.9 10.6 12.4 14.2 16.0 8.5 10.2 11.9 13.6 15.3 9.811.5 13.1 14.8 8.2 7.9 9.5 11.1 12.6 14.2 9.110.7 12.2 13.7 8.8 10.3 11.8 13.3 36 37 315 328 340 353 366 378 391 403 416 428 35 5441 453 465 478 490 502 514 527 539 5513 563 575 587 599 611 623 635 647 658 670 682 694 705 717 729 740 752 763 775 786 1.3 2.5 3.8 5.0 6.3 7.6 8.8 10.1 11.3 2.4 3.6 4.8 5.9 7.1 8.3 9.5 10.7 2.3 3.5 4.6 5.8 6.9 8.1 9.3 10.4 1.1 2.3 3.4 4.5 5.6 6.8 79 9.0 10.2 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 2. I 3.2 4.3 5.4 6.4 7.5 8.6 9.7 2.1 3.1 4.2 5.2 6.3 7.3 8.4 9.4 2.0 3.1 4.1 5.1 6.1 7.2 8.2 9.2 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 2.0 2.9 3.9 4.9 5.9 6.8 7.8 8.8 1.9 2.9 3.8 4.8 5.7 6.7 7.6 8.6 0.9 1.9. 2.8 3.7 4.7 5.6 6.5 7 5 8.4 0.9 1.8 2.7 3.7 4.6 5.5 6.4 7.3 8.2 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 1.8 2.6 3.5 4.4 5.3 6.1 7.0 7.9 0.9 1.7 2.6 3.4 4.3 5.2 6.0 6.9 7.7 0.8 1.7 2.5 3.4 4.2 5.1 5.9 6.7 7.6 1.0 40 6021 031 042 053 064 075 085 096 107 117 66 67 73 65 8129 136 142 149 156 162 169 176 182 189 195 202 209 215 222 228 235 241 248 254 261 267 274 280 287 293 299 306 312 319 68 325 331 338 344 351 357 363 370 376 382 69 388 395 401 407 414 420 426 432 439 445 70 8451 457 463 470 476 482 488 494 500 506 71 513 519 525 531 537 543 549 555 561 567 72 573 579 585 591 597 603 609 615 621 627 633 639 645 651 657 663 669 675 681 686 74 692 698 704 710 716 722 727 733 739 745 75 8751 756 762 768 774 779 785 791 797 802 808 814 820 825 831 837 842 848 854 859 77 865 871 876 882 887 893 899 904 910 915 78 921 927 932 938 943 949 954 960 965 971 79 976 982 987 993 998 004 009 015 20 25 80 9031 036 042 047 053 058 063 069 074 079 81 085 090 096 101 106 112 117 122 128 133 82 138 143 149 154 159 165 170 175 180 186 191 196 201 206 212 217 222 227 232 238 243 248 253 258 263 269 274 279 284 289 76 83 84 86 87 88 91 92 93 85 9294 299 304 309 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 469 474 479 484 489 89 494 499 504 509 513 518 523 528 533 538 90 9542 547 552 557 562 566 571 576 581 586 590 595 600 605 609 614 619 624 628 633 638 643 647 652 657 661 666 671 675 680 685 689 694 699 703 708 713 717 722 727 94 731 736 741 745 750 754 759 763 768 773 95 9777 782 786 791 795 800 805 809 814 818 823 827 832 836 841 845 850 854 859 863 868 872 877 881 886 890 894 899 903 908 912 917 921 926 930 934 939 943 948 952 99 956 961 965 969 974 978 983 987 991 996 96 97 98 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 2.7 3-4 4.1 4.8 5.5 6.2 1.3 2.0 2.7 3.4 4.0 4.7 5.4 6.1 0.7 0.7 0.7 1.3 2.0 2.7 3.3 4.0 4.6 5.3 6.0 0.7 1.3 2.0 2.6 3.3 3.9 4.6 5.2 5.9 0.6 1.3 1.9 2.6 3.2 3.9 4.5 5.1 5.8 0.6 1.3 1.9 2.5 3.2 3.8 4.4 5.1 5.7 0.6 1.2 1.9 2.5 3.1 3.7 4.4 5.0 5.6 0.6 1.2 1.8 2.5 3.1 3.7 4.3 4.9 5.5 0.6 1.2 1.8 2.4 3.0 3.6 4.3 4.9 5.5 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 0.6 1.2 1.8 2.4 3.0 3.5 4.1 4.7 5.3 0.6 1.2 1.7 2.3 2.9 3.5 4.1 4.7 5.2 0.6 1.2 1.7 2.3 2.9 3.5 4.0 4.6 5.2 0.6 1.7 2.3 2.8 3.4 4.0 4.5 5.1 0.6 I.I 1.7 2.2 2.8 3.4 3.9 4.5 5.0 0.6 1.1 1.7 2.2 2.8 3.3 3.9 4.4 5.0 0.5 1.1 1.6 2.2 2.7 3.3 3.8 4.4 4.9 0.5 1.1 1.6 2.2 2.7 3.2 3.8 4.3 4.9 0.5 1.1 1.6 2.1 2.7 3.2 3.7 4.3 4.8 I. I 1.6 2.1 2.6 3.2 3.7 4.2 4.7 0.5 1.0 1.6 2.1 2.6 3.1 3.6 4.2 4.7 0.5 1.0 1.5 2.1 2.6 3.1 3.6 4.1 4.6 0.5 0.5 0.5 I.I 1.0 1.5 2.0 2.5 3.0 3.6 4.4 4.6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.5 1.O 1.5 2.0 2.5 2.9 3.4 3.9 4.4 0.5 1.0 1.5 1.9 2.4 29 3.4 3.9 4.4 0.5 1.0 I.4 1.9 2.4 2.9 3.4 3.8 4.3 0.5 0.9 1.4 1.9 2.4 2.8 3.3 3.8 4.3 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.8 4.2 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 0.5 0.9 1.4 1.8 2.3 2.8 3.2 3.7 4.1 0.5 0.9 1.4 1.8 2.3 2.7 3.2 3.6 4.1 0.5 0.9 1.4 1.8 2.3 2.7 3.2 3.6 4.1 0.4 0.9 1.3 1.8 2.2 2.7 3.1 3.6 4.0 0.4 0.9 1.3 1.8 2.2 2.6 3.1 3.5 4.0 0.4 0.9 1.3 1.7 2.2 2.6 3.1 3.5 3.9 |