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Logarithms

This page serves as a comprehensive resource providing essential logarithm tables. Specifically, it includes values for both common logarithms, typically denoted as $\log$ and based on $\mathbf{10}$, and natural logarithms, often written as $\ln$ and based on the mathematical constant $\mathbf{e}$ (approximately $\approx 2.71828$). These tables have been fundamental tools throughout the history of computation, offering a direct reference for these critical mathematical values.

At their core, logarithms are fundamentally exponents. Understanding this relationship is key: the logarithm of a number 'y' to a given base 'b' is simply the exponent 'x' to which the base 'b' must be raised to produce 'y'. This fundamental relationship can be expressed mathematically as: if $b^x = y$, then $\log_b(y) = x$. The tables presented here provide pre-calculated 'x' values for a wide range of 'y' values, based on the specified 'b' (either 10 or e).

Historically, before the widespread availability of electronic calculators, logarithm tables were indispensable tools for scientists, engineers, and anyone performing complex calculations. They offered a revolutionary method for simplifying arithmetic operations, particularly multiplication and division, which were labor-intensive and prone to errors with large numbers. By converting numbers to their logarithmic equivalents using the tables, complex problems could be transformed into simpler additions and subtractions, leveraging the powerful properties of logarithms:

Using these tables involved finding the logarithm of each number, performing the simplified addition or subtraction, and then using the tables in reverse (finding the antilogarithm) to convert the result back. Tables were structured in a grid format, allowing users to locate values by aligning rows and columns based on the number's digits. This process, while seemingly cumbersome today, represented a massive efficiency gain compared to manual calculation of large numbers.

While modern calculators now perform these logarithmic and antilogarithmic operations instantly and with high precision, the skill of using logarithm tables remains valuable for educational purposes. It provides crucial insight into the nature of logarithmic scales, the historical development of computational tools, and the clever mathematical principles that underpin logarithmic operations. Furthermore, understanding these tables helps in grasping the concept of magnitude changes that are represented by logarithmic scales.

Logarithmic scales are prevalent in various scientific and engineering disciplines, used whenever quantities vary over enormous ranges. Familiar examples include:

This page serves as a direct reference for these fundamental logarithmic values, preserving the legacy of historical computation while providing a practical resource for educational and conceptual understanding.

0 1 2 3 4 5 6 7 8 9
10 0000 0043 0086 0128 0170
0212 0253 0294 0334 0374
11 0414 0453 0492 0531 0569
0607 0645 0682 0719 0755
12 0792 0828 0864 0899 0934 0969
1004 1038 1072 1106
13 1139 1173 1206 1239 1271
1303 1335 1367 1399 1430
14 1461 1492 1523 1553
1584 1614 1644 1673 1703 1732
15 1761 1790 1818 1847 1875 1903
1931 1959 1987 2014
16 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
1 2 3 4 5 6 7 8 9
4 9 13 17 21 26 30 34 38
4 8 12 16 20 24 28 32 37
4 8 12 15 19 23 27 31 35
4 7 11 15 19 22 26 30 33
3 7 11 14 18 21 25 28 32
3 7 10 13 16 20 23 26 30
3 7 10 13 16 20 23 26 30
3 7 10 12 16 19 22 25 29
3 6 9 12 15 18 21 24 28
3 6 9 12 15 17 20 23 26
3 6 9 11 14 17 20 23 26
3 5 8 10 13 15 18 21 23
3 5 8 11 14 16 19 22 24
3 5 8 10 13 15 18 21 23
3 5 8 10 13 15 18 20 23
2 5 7 10 12 15 17 19 22
2 5 7 9 12 14 16 19 21
2 5 7 9 11 14 16 18 21
2 4 7 9 11 13 16 18 20
2 4 6 8 11 13 15 17 19
0 1 2 3 4 5 6 7 8 9
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 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133
26 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
1 2 3 4 5 6 7 8 9
2 4 6 8 11 13 15 17 19
2 4 6 8 10 12 14 16 18
2 6 6 8 10 12 14 15 17
2 4 6 7 9 11 13 15 17
2 4 5 7 9 11 12 14 16
2 3 5 7 9 10 12 14 15
2 3 7 7 8 10 11 13 15
2 3 5 6 8 9 11 13 14
2 3 5 6 8 9 11 12 14
1 3 4 6 7 9 10 12 13
1 3 4 6 7 9 10 11 31
0 1 2 3 4 5 6 7 8 9
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
40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117
1 2 3 4 5 6 7 8 9
1 3 4 6 7 8 10 11 12
1 3 4 5 7 8 9 11 12
1 3 4 5 6 8 9 10 12
1 3 4 5 6 8 9 10 11
1 2 4 5 6 7 9 10 11
1 2 4 5 6 7 8 10 11
1 2 3 5 6 7 8 9 10
1 2 3 5 6 7 8 9 10
1 2 3 4 5 7 8 9 10
1 2 3 4 5 6 8 9 10
0 1 2 3 4 5 6 7 8 9
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 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618
46 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
49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981
50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067
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 7 8
1 2 3 4 5 5 6 7 8
1 2 3 4 4 5 6 7 8
1 2 3 4 4 5 6 7 8
1 2 3 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8 9
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 7293 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
1 2 3 4 5 6 7 8 9
1 2 3 3 4 5 6 7 8
1 2 2 3 4 5 6 7 7
1 2 2 3 4 5 6 6 7
1 2 2 3 4 5 6 6 7
1 2 2 3 4 5 5 6 7
1 2 2 3 4 5 5 6 7
1 2 2 3 4 5 5 6 7
1 1 2 3 4 4 5 6 7
1 1 2 3 4 4 5 6 7
1 1 2 3 4 4 5 6 6
0 1 2 3 4 5 6 7 8 9
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
70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506
1 2 3 4 5 6 7 8 9
1 1 2 3 4 4 5 6 6
1 1 2 3 3 4 5 6 5
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 3 4 5 5 6
1 1 2 3 3 4 5 5 6
1 1 2 3 3 4 4 5 6
1 1 2 2 3 4 4 5 6
1 1 2 2 3 4 4 5 6
0 1 2 3 4 5 6 7 8 9
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
75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802
76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859
77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915
78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971
79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025
80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079
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 5
1 1 2 2 3 4 4 5 5
1 1 2 2 3 4 4 5 5
1 2 2 2 3 3 4 5 5
1 1 2 2 3 3 4 5 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
0 1 2 3 4 5 6 7 8 9
81 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
84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289
85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340
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
1 2 3 4 5 6 7 8 9
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 4 5
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 1 2 2 3 3 4 4
0 1 2 3 4 5 6 7 8 9
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
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
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 1 2 2 3 3 4 4
0 1 1 2 2 3 3 4 4
0 1 1 2 2 3 3 3 4