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Antilogarithms

This page is specifically dedicated to providing antilogarithm tables. These tables perform the crucial inverse operation to finding a logarithm. Whereas a logarithm table answers the question "To what power must the base be raised to get this number?", an antilogarithm table answers the question "What number do you get when the base is raised to this power?".

In simpler terms, an antilogarithm essentially "undoes" the process of taking a logarithm. If you know that the logarithm of some unknown number 'x' to a base 'b' is 'y' (i.e., $\log_b(x) = y$), then the antilogarithm of 'y' with respect to the same base 'b' will give you the original number 'x'. This inverse relationship can be elegantly expressed mathematically: if $\log_b(x) = y$, then the antilogarithm of 'y' (base 'b') is $x = b^y$. The tables presented here allow you to find this value of 'x' when you are given the value of 'y'.

Historically, antilogarithm tables were just as critical as logarithm tables themselves, forming the second necessary step in many complex calculations. When using logarithms to simplify multiplication or division, you would first convert the original numbers to their logarithms using logarithm tables, perform the addition or subtraction, and then use the antilogarithm table to convert the resulting logarithmic sum or difference back into the final numerical answer. They were the essential bridge returning from the simplified logarithmic domain back to the standard numerical scale.

Using an antilogarithm table typically involves looking up a given logarithmic value (the 'y' in the equation $\log_b(x) = y$) within the table's structure. When dealing with common logarithms (base 10), the lookup process usually involves considering both the characteristic (the integer part of the logarithm) and the mantissa (the fractional part). You would use the mantissa to navigate the rows and columns of the table to find a sequence of digits, and the characteristic would then tell you where to place the decimal point in that sequence to obtain the correct antilogarithm value.

It is absolutely essential that the base of the antilogarithm table you use matches the base of the logarithm that was originally taken. For example, if you calculated a logarithm using a common logarithm table (base 10), you must use a common antilogarithm table (also base 10) to find the original number. Attempting to use a table with a different base would yield an incorrect result.

While electronic calculators and computers have largely superseded the manual use of these tables for everyday computation, antilogarithm tables retain significant value. They are invaluable resources for educational purposes, helping students understand the exponential relationship that is the inverse of the logarithmic relationship. They also provide insight into the complete historical process of using logarithms for calculation, illustrating the ingenuity required to perform complex arithmetic before modern technology. This resource provides the necessary data to perform this inverse operation directly.

0 1 2 3 4 5 6 7 8 9
00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021
.01 1023 1026 1028 1030 1033 1035 1038 1040 1042 1045
.02 1047 1050 1052 1054 1057 1059 1062 1064 1067 1069
.03 1072 1074 1076 1079 1081 1084 1086 1089 1091 1094
.04 1096 1099 1102 1104 1107 1109 1112 1114 1117 1119
.05 1122 1125 1127 1130 1132 1135 1138 1140 1143 1146
.06 1148 1151 1153 1156 1159 1161 1164 1167 1169 1172
.07 1175 1178 1180 1183 1186 1189 1191 1194 1197 1199
.08 1202 1205 1208 1211 1213 1216 1219 1222 1225 1227
.09 1230 1233 1236 1239 1242 1245 1247 1250 1253 1256
1 2 3 4 5 6 7 8 9
0 0 1 1 1 1 2 2 2
0 0 1 1 1 1 2 2 2
0 0 1 1 1 1 2 2 2
0 0 1 1 1 1 2 2 2
0 1 1 1 1 2 2 2 2
0 1 1 1 1 2 2 2 2
0 1 1 1 1 2 2 2 2
0 1 1 1 1 2 2 2 2
0 1 1 1 1 2 2 2 3
0 1 1 1 1 2 2 2 3
0 1 2 3 4 5 6 7 8 9
.10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285
.11 1288 1291 1294 1297 1300 1303 1306 1309 1312 1315
.12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346
.13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377
.14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409
.15 1413 1416 1419 1422 1426 1429 1432 1435 1439 1442
.16 1445 1449 1452 1455 1459 1462 1466 1469 1472 1476
.17 1479 1483 1486 1489 1493 1496 1500 1503 1507 1510
.18 1514 1517 1521 1524 1528 1531 1535 1538 1542 1545
.19 1549 1552 1556 1560 1563 1567 1570 1574 1578 1581
.20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618
1 2 3 4 5 6 7 8 9
0 1 1 1 1 2 2 2 3
0 1 1 1 2 2 2 2 3
0 1 1 1 2 2 2 2 3
0 1 1 1 2 2 2 3 3
0 1 1 1 2 2 2 3 3
0 1 1 1 2 2 2 3 3
0 1 1 1 2 2 2 3 3
0 1 1 1 2 2 2 3 3
0 1 1 1 2 2 2 3 3
0 1 1 1 2 2 3 3 3
0 1 1 1 2 2 3 3 3
0 1 2 3 4 5 6 7 8 9
.21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656
.22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694
.23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734
.24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774
.25 1778 1782 1786 1791 1795 1799 1803 1807 1811 1816
.26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858
.27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901
.28 1905 1910 1914 1919 1923 1928 1932 1936 1941 1945
.29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991
.30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037
1 2 3 4 5 6 7 8 9
0 1 1 2 2 2 3 3 3
0 1 1 2 2 2 3 3 3
0 1 1 2 2 2 3 3 4
0 1 1 2 2 2 3 3 4
0 1 1 2 2 2 3 3 4
0 1 1 2 2 3 3 3 4
0 1 1 2 2 3 3 3 4
0 1 1 2 2 3 3 4 4
0 1 1 2 2 3 3 4 4
0 1 1 2 2 3 3 4 4
0 1 2 3 4 5 6 7 8 9
.31 2042 2046 2051 2056 2061 2065 2070 2075 2080 2084
.32 2089 2094 2099 2104 2109 2113 2118 2123 2128 2133
.33 2138 2143 2148 2153 2158 2163 2168 2173 2178 2183
.34 2188 2193 2198 2203 2208 2213 2218 2223 2228 2234
.35 2239 2244 2249 2254 2259 2265 2270 2275 2280 2286
.36 2291 2296 2301 2307 2312 2317 2323 2328 2333 2339
.37 2344 2350 2355 2360 2366 2371 2377 2382 2388 2393
.38 2399 2404 2410 2415 2421 2427 2432 2438 2443 2449
.39 2455 2460 2466 2472 2477 2483 2489 2495 2500 2506
.40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564
1 2 3 4 5 6 7 8 9
0 1 1 2 2 3 3 4 4
0 1 1 2 2 3 3 4 4
0 1 1 2 2 3 3 4 4
1 1 2 2 3 3 4 4 5
1 1 2 2 3 3 4 4 5
1 1 2 2 3 3 4 4 5
1 1 2 2 3 3 4 4 5
1 1 2 2 3 3 4 4 5
1 1 2 2 3 3 4 5 5
1 1 2 2 3 4 4 5 5
0 1 2 3 4 5 6 7 8 9
.41 2570 2576 2582 2582 2588 2594 2600 2606 2612 2624
.42 2630 2636 2642 2649 2655 2661 2667 2673 2679 2685
.43 2692 2698 2704 2710 2716 2723 2729 2735 2742 2748
.44 2754 2761 2767 2773 2780 2786 2793 2799 2805 2812
.45 2818 2825 2831 2838 2844 2851 2858 2864 2871 2877
.46 2884 2891 2897 2904 2911 2917 2924 2931 2938 2944
.47 2951 2958 2965 2972 2979 2985 2992 2999 3006 3013
.48 3020 3027 3034 3041 3048 3055 3062 3069 3076 3083
.49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155
.50 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228
1 2 3 4 5 6 7 8 9
1 1 2 2 3 4 4 5 5
1 1 2 2 3 4 4 5 6
1 1 2 3 3 4 4 5 6
1 1 2 3 3 4 4 5 6
1 1 2 3 3 4 5 5 6
1 1 2 3 3 4 5 5 6
1 1 2 3 3 4 5 5 6
1 1 2 3 4 4 5 6 6
1 1 2 3 4 4 5 6 6
1 1 2 3 4 4 5 6 7
0 1 2 3 4 5 6 7 8 9
.51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304
.52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381
.53 3388 3396 3404 3412 3420 3428 3436 3443 3451 3459
.54 3467 3475 3483 3491 3499 3508 3516 3524 3532 3540
.55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622
.56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707
.57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793
.58 3802 3811 3819 3828 3837 3846 3855 3864 3873 3882
.59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972
.60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064
1 2 3 4 5 6 7 8 9
1 1 2 3 4 4 5 6 6
1 2 2 3 4 5 5 6 7
1 2 2 3 4 5 5 6 7
1 2 2 3 4 5 6 6 7
1 2 2 3 4 5 6 7 7
1 2 3 3 4 5 6 7 8
1 2 3 3 4 5 6 7 8
1 2 3 4 4 5 6 7 8
1 2 3 4 5 5 6 7 8
1 2 3 4 5 6 6 7 8
0 1 2 3 4 5 6 7 8 9
.61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4159
.62 4169 4178 4188 4198 4207 4217 4227 4236 4246 4256
.63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355
.64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457
.65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560
.66 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667
.67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775
.68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887
.69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000
.70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 9 10
1 2 3 4 5 7 8 9 10
1 2 3 4 6 7 8 9 10
1 2 3 5 6 7 8 9 10
1 2 4 5 6 7 8 9 11
0 1 2 3 4 5 6 7 8 9
.71 5129 5140 5152 5164 5176 5188 5200 5212 5224 5236
.72 5248 5260 5272 5284 8297 5309 5321 5333 5346 5358
.73 5370 5383 5395 5408 5420 5433 5445 5458 5470 5483
.74 5495 5508 5521 5534 5546 5559 5572 5585 5598 5610
.75 5623 5636 5649 5662 5675 5689 5702 5715 5728 5741
.76 5754 5768 5781 5794 5808 5821 5834 5848 5861 5875
.77 5888 5902 5916 5929 5943 5957 5970 5984 5998 6012
.78 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152
.79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295
.80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442
1 2 3 4 5 6 7 8 9
1 2 4 5 6 7 8 10 11
1 2 4 5 6 7 9 10 11
1 3 4 5 6 8 9 10 11
1 3 4 5 6 8 9 10 12
1 3 4 5 7 8 9 10 12
1 3 4 5 7 8 9 11 12
1 3 4 5 7 8 10 11 12
1 3 4 6 7 8 10 11 13
1 3 4 6 7 9 10 11 13
1 3 4 6 7 9 10 12 13
0 1 2 3 4 5 6 7 8 9
.81 6457 6471 6486 6501 6516 6531 6546 6561 6577 6592
.82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745
.83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902
.84 6918 6934 6950 6966 6982 6998 7015 7031 7047 7063
.85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228
.86 7244 7261 7278 7295 7311 7328 7345 7362 7379 7396
.87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568
.88 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745
.89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925
.90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110
1 2 3 4 5 6 7 8 9
1 3 5 6 8 9 11 12 14
2 3 5 6 8 9 11 12 14
2 3 5 6 8 9 11 13 14
2 3 5 6 8 10 11 13 15
2 3 5 7 8 10 12 13 15
2 3 5 7 8 10 12 13 15
2 3 5 7 9 10 12 14 16
2 4 5 7 9 11 12 14 16
2 4 5 7 9 11 13 14 16
2 4 6 7 9 11 13 15 17
0 1 2 3 4 5 6 7 8 9
.91 8128 8147 8166 8155 8204 8222 8241 8260 8279 8299
.92 8313 8337 8356 8375 8395 8414 8433 8453 8472 8492
.93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690
.94 8710 8730 8750 8770 8790 8810 8831 8851 8872 8892
.95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099
.96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311
.97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528
.98 9550 9572 9594 9616 9638 9661 9683 9705 9727 9750
.99 9772 9795 9817 9840 9863 9886 9908 9931 9954 9977
1 2 3 4 5 6 7 8 9
2 4 6 8 9 11 13 15 17
2 4 6 8 10 12 14 15 17
2 4 6 8 10 12 14 16 18
2 4 6 8 10 12 14 16 18
2 4 6 8 10 12 15 17 19
2 4 6 8 11 13 15 17 19
2 4 7 9 11 13 15 17 20
2 4 7 9 11 13 16 18 20
2 5 7 9 11 14 16 18 20