TABLE I LOGARITHMS OF NUMBERS. EXPLANATION. LOGARITHMS OGARITHMS are a series of numbers so contrived, that the sum of the Logarithms of any two numbers, is the logarithm of the product of these numbers. Hence it is inferred, that if a rank, or series of numbers in arithmetical progression, be adapted to a series of numbers in geometrical progression, any term in the arithmetical progression will be the logarithm of the corresponding term in the geometrical progression. This table contains the common logarithms of all the natural numbers from 0 to 10000, calculated to six decimal places; such, on account of their superior accuracy, being preferable to those, that are computed only to five places of decimals. In this form, the logarithm of 1 is 0, of 10, 1; of 100, 2; of 1000, 3 &c. Whence the logarithm of any term between 1 and 10, being greater than O, but less than 1, is a proper fraction, and is expressed decimally. The logarithm of each term between 10 and 100, is 1, with a decimal fraction annexed; the logarithm of each term between 100 and 1000 is 2, with a decimal annexed, and so on. The integral part of the logarithm is called the Index, and the other the decimal part.-. Except in the first hundred logarithms of this Table, the Indexes are not printed, being so readily supplied by the operator from this general rule; the Index of a Logarithm is always one less than the number of figures contained in its corresponding natural number-exclusive of fractions, when there are any in that number. The Index of the logarithm of a number, consisting in whole, or in parts, of integers, is affirmative; but when the value of a number is less than unity, or 1, the index is negative, and is usually marked by the sign,, placed either before, or above the index. If the first significant figure of the decimal fraction be adjacent to the decimal point, the index is 1, or its arithmetical complement 9; if there is one cipher between the decimal point and the first significant figure in the decimal, the index is. 2, or its arith. comp. 8; if two ciphers, the index is 3, or 7, and so on; but the arithmetical complements, 9, 8, 7 &c. are rather more conveniently used in trigonometrical calculations, - A The decimal parts of the logarithms of numbers, consisting of the same figures, are the same, whether the number be integral, fractional, or mixed thus, N. B. The arithmetical complement of the logarithm of any number, is found by subtracting the given logarithm from that of the radius, or by subtracting each of its figures from 9, except the last, or right-hand figure, which is to be taken from 10. The arithmetical complement of an index is found by subtracting it from 10. PROBLEM I. To find the logarithm of any given number. RULES. 1. If the number is under 100, its logarithm is found in the first page of the table, immediately opposite thereto. Thus the Log. of 53, is 1.724276. 2. If the number consists of three figures, find it in the first column of the following part of the table, opposite to which, and under 0, is its logarithm. Thus the Log, of 384 is 2.584331-prefixing the index 2, because the natural number contains 3 figures. Again the log. of 65.7 is 1.817565-prefixing the index 1, because there are two figures only in the integral part of the given number. 3. If the given number contains four figures, the three first are to be found, as before, in the side column, and under the fourth at the top of the table is the logarithm required. Thus the log. of 8735 is 3.941263-for against 873, the three first figures found in the left side column, and under 5, the fourth figure found at the top, stands the decimal part of the logarithm, viz .941263, to which prefixing the index, 3, because there are four figures in the natural number, the proper logarithm is obtained. Again the logarithm of 37.68 is 1.576111-Here the decimal part of the logarithm is found, as before, for the four figures; but the index is 1, because there are two integral places only in the natural number. 4. If the given number exceeds four figures, find the difference between the logarithms answering to the first four figures of the given number, and the next following logarithm; multiply this difference by the remaining figures in the given number, point off as many figures to the right-hand as there are in the multiplier, and the remainder, add J ed to the logarithm, answering to the first four figures, will be the re quired logarithm, nearly. Thus; to find the logarithm of 738582; the log. of the first four figures, viz. 7385 the next greater logarithm to be multiplied by the remaining figures .868350 = 868409 = 82 118 472 48|38 the sum 5.868398, with the proper index prefixed, is the required logarithm. 5. The logarithm of a vulgar-fraction is found by subtracting the logarithm of the denominator from that of the numerator; and that of a mixed quantity is found by reducing it to an improper fraction, and proceeding as before. Thus to find the Logarithm of 7; from the log. of 7 subtract the log. of 8 Remainder = 0.845098 = 0.903090 9.942008 the required log. PROBLEM II. To find the number answering to any given logarithm. RULES. 1. Find the next less logarithm to that given in the column marked o at the top, and continue the sight along that horizontal line, and a logarithm the same as that given, or very near it, will be found; then the three first figures of the corresponding natural number will be found opposite thereto in the side column, and the fourth figure immediately above it, at the top of the page. If the index of the given logarithm is 3, the four figures thus found are integers; if the index is 2, the three first figures are integers, and the fourth is a decimal, and so on. Thus the log. 3.132580 gives the Nat. Numb. 1357 2.132580 gives 135.7 13.57 1.357 .1357 &c. 2. If the given logarithm cannot be exactly found in the table, and if more than four figures be wanted in the corresponding natural number; then find the difference between the given and the next less loga rithms, to which annex as many ciphers as there are figures required above four in the natural number; which divide by the difference between the next less, and next greater logarithms, and the quotient an nexed to the four figures formerly found, will give the required natural number. Thus to find the natural number of the log. 4.828991; the next less log. is .828982 which gives 6735; the next greater log. is 829046 therefore 1.4 being annexed to 6735, the required natural number, 67351.4, is now obtained. TABLE I. LOGARITHMS OF NUMBERS. No. Log. Log. No. Log. No. Log. No. Log. No. Lo No. 0 100 3 4 5 6 7 8 9 2 000000000434 000868 001301 001734 002166 002598003029003460003891 101 004321004751 005180 005609 006038 006466 006894 007321 007748 008174 102 008600 009206 009451009876 010300 010724 011147 011570 011993 012415 103 012837 013259 013680 014100 014520014940 015360 015779 016197 016615 104 017033 017451 017868 018284 018700 019116 019532 019947 020361 020775 105 021189 021603 022016 022428 022841 023252023664024075 024486 024896 106 025306 025715.026124 026533 026942 027350 027757028164 028571 028978 107 029384 029789 030195 030600 031004 031408 031812 032216 032619 033021 108 0334241033826 034227 034628 035029|035430035830036229|036629|037028 109 037426.037825 038223 038620 039017 039414039811|040207 040602|040998 110 041393 041787 042182 042575 042969 043362 043755 044148044540044931 111 045323 045714 046105 046495 046885 047275 047664 048053 C442 048830 112 049218 049606 049993 050380 050766 051152 051538 051924 052309 052694 113 053078053463 053846 054230 054613 054996 055378 055760 056142 056524 114056905057286 057666 058046 058426 058805 059185 059563 059942 060320 115 060698 061075 061452 061829 062206 062582 062958 063333 063709 064083 116 064458 064832 065206 065580065953 066326 066699|067071 067443 067814 117068186 068557 068928 069298 069668 070038 070407 070776 071145 071514 118 071882 072250 072617 072985 073352 073718 074085 074451 074816075182 119 075547 075912 076276 076640 077004 077368 077731 078094 078457 078819 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 121 082785 083144 083503 083861 084219 084576 084934 085291 085647 086004 122 086360 086716 087071 087426 087781 088136 088490 088845 089198 089552 123 089905 090258 090611 090963 091315 091667 092018 092370 092721 093071 124 093422 093772 094122 094471 094820 095169 095518095366 096215 096562 125 096910 097257 097604 097951 098297 098644 098990099 335 099681 100026 126 100370 100715 101059 101403 101747 102090 102434 102777 103119 103462 127 103804 104146 104487 104828 105169 105510 105851 106191 106531 106870 128 107210 107549 107888 108227 108565 108903 109241 109578 109916 110253 129 110590 110926 111262 111598 111934 112270112605112940 113275 113609 130113943 114277 114611 114944 115278 115610 115943 116276 116608 116940 131 117271 117603 117934 118265 118595 118926 119256 119586 119915 120245 132 120574 120903 121231 121560121888 122216 122543 122871 123198 123525 133123852 124178 124504 124830 125156 125481| 125806 126131 126456 126781 134 127105 127429 127752 128076 128399 128722 129045 129368 129690 130012 135130334 130655 130977 131298 131619 131939 132260132580 132900 133219 136 133539 133858 134177 134496 134814135133 135451 135768 136086 136403 137 136721 137037137354 137670 137987 138303 138618 138934139249 139564 138 139879 140194 140508 140822 141136 141450 141763 142076 142389 142702 139 143015 143327 143639 143951 144263 144574 144885145196 145507 145818 140 146128 146438 146748 147058 147367 147676 147985 148294 148603 148911 141149219 149527 149835 150142 150449 150756 151063 151370 151676 151982 142 152288 152594 152900 153205 153510153815 154119 154424 154728 155032 143 155336 155640 155943 156246 156549 156852157154 157457 157759 158061 144 158362 158664 158965 159266 159567 159868 160168 160468 160769 161068 145 161368 161667 161967 162266 162564162863 163161 163460 163757 164055 146 164353 164650 164947 165244 165541 165838 166134 166430 166726 167022 147167317 167613 167908 168203 168497 168792 169086 169380 169674 169968 148 170262 170555 170848 171141 171434 171726 172019 172311172603 172895 149 173186 173478 173769 174060 174351 174641 174932 175222 175512 175802 150 176091 176381 176670 176959 177248 177536 177825 178113 178401 178689 151 178977 179264 179552 179839 180126 180413 180699 180986 181272 181558 152 181844 182129 182415 182700 182985 183270 183554 183839 184123 184407 153 184691184975 185259 185542 185825 186108 186391 186674 186956 187239 154 187521 187803 188084 188366 188647 188928 189209 189490 189771 190051 155190332 190612 190892 191 171 191451 19 1730 1920 10 192289 192567|192846 156 193125 193403 193681 193959 194237 194514 194792 195069 195346 195623 157195900 196176 196452 196729 197005197281 197556 197832 198107 198382 158 198657 198932 199206|199481199755 200029 200303 200577 2008 50 201124 159 201397 201670 201943 202216 202488 202761 203033 203305 203577 203848 |