1422:
818:
122:
1417:{\displaystyle {\begin{array}{rll}{\text{As found above,}}&\log _{10}(0.012)\approx {\bar {2}}.07918\\{\text{Since}}\;\;\log _{10}(0.85)&=\log _{10}\left(10^{-1}\times 8.5\right)=-1+\log _{10}(8.5)&\approx -1+0.92942={\bar {1}}.92942\\\log _{10}(0.012\times 0.85)&=\log _{10}(0.012)+\log _{10}(0.85)&\approx {\bar {2}}.07918+{\bar {1}}.92942\\&=(-2+0.07918)+(-1+0.92942)&=-(2+1)+(0.07918+0.92942)\\&=-3+1.00860&=-2+0.00860\;^{*}\\&\approx \log _{10}\left(10^{-2}\right)+\log _{10}(1.02)&=\log _{10}(0.01\times 1.02)\\&=\log _{10}(0.0102).\end{array}}}
259:
1866:
25:
1980:
3023:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications.
592:
597:
To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, one can express a negative logarithm as a negative integer characteristic plus a positive mantissa. To facilitate this, a special notation, called
440:
The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is 2.
1962:
for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes
1782:
435:
298:
An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the
1873:
scales at distances proportional to the differences between their logarithms. By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that
310:, can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by the following calculation:
2273:
2177:
672:
1913:) logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first
278:. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions. Because logarithms were so useful,
2081:
804:
2506:
2422:
303:. Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999.
455:
1638:
1971:" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.
2791:
316:
2748:
2683:
2715:
740:
1828:
704:
262:
Page from a table of common logarithms. This page shows the logarithms for numbers from 1000 to 1509 to five decimal places. The complete table covers values up to 9999.
2303:
1848:
1630:
677:
The bar over the characteristic indicates that it is negative, while the mantissa remains positive. When reading a number in bar notation out loud, the symbol
2182:
2086:
1995:) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation.
274:
of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a
282:
of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of
608:
2992:
2005:
748:
2427:
2343:
89:
3098:
3073:
3028:
61:
2854:
587:{\displaystyle \log _{10}(0.012)=\log _{10}\left(10^{-2}\times 1.2\right)=-2+\log _{10}(1.2)\approx -2+0.07918=-1.92082.}
68:
108:
42:
809:
with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.
742:
is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case
209:(logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log". To mitigate this ambiguity, the
1777:{\displaystyle \log _{10}\left(x\times 10^{i}\right)=\log _{10}(x)+\log _{10}\left(10^{i}\right)=\log _{10}(x)+i.}
75:
2583:
stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text. Nowadays, the word
1434:
The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten:
46:
57:
2315:
1898:
1862:, 0.698 970 (004 336 018 ...) will be listed once indexed by 5 (or 0.5, or 500, etc.).
430:{\displaystyle \log _{10}(120)=\log _{10}\left(10^{2}\times 1.2\right)=2+\log _{10}(1.2)\approx 2+0.07918.}
154:
2818:
2757:
2720:
2656:
1999:
The numerical value for logarithm to the base 10 can be calculated with the following identities:
2688:
713:
2897:
Institutionum
Analyticarum Pars Secunda de Calculo Infinitesimali Liber Secundus de Calculo Integrali
2880:
2609:(1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen".
1794:
35:
1920:
Because base-10 logarithms were most useful for computations, engineers generally simply wrote "
680:
2824:
2817:
Hall, Arthur Graham; Frink, Fred
Goodrich (1909). "Chapter IV. Logarithms Common logarithms".
283:
3063:
82:
3018:
2567:
2323:
1892:
267:
3084:
3046:
2984:
2970:
2628:
2281:
8:
3117:
2964:
2537:
2632:
2868:
2644:
2618:
1833:
1615:
279:
271:
3094:
3069:
3050:
3034:
3024:
3010:
2939:
2911:
2648:
2532:
2308:
812:
The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:
206:
823:
2636:
2606:
2517:
2319:
270:
capable of multiplication were bulky, expensive and not widely available. Instead,
258:
3042:
3020:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
2895:
2850:
2846:
2842:
266:
Before the early 1970s, handheld electronic calculators were not available, and
2838:
2588:
2268:{\displaystyle \quad \log _{10}(x)={\frac {\log _{B}(x)}{\log _{B}(10)}}\quad }
2172:{\displaystyle \quad \log _{10}(x)={\frac {\log _{2}(x)}{\log _{2}(10)}}\quad }
1901:, a 17th century British mathematician. In 1616 and 1617, Briggs visited
1438:
Common logarithm, characteristic, and mantissa of powers of 10 times a number
3111:
2640:
121:
3083:
Poliyanin, Andrei
Dmitrievich; Manzhirov, Alexander Vladimirovich (2007) .
3014:
2860:
2592:
2542:
2522:
1902:
1897:
Common logarithms are sometimes also called "Briggsian logarithms" after
1610:
210:
130:
1870:
275:
202:
2900:(in Latin). Vol. 2. Joannis Thomæ Nob. De Trattnern. p. 198.
1865:
3090:
1906:
1851:
667:{\displaystyle \log _{10}(0.012)\approx {\bar {2}}+0.07918=-1.92082.}
287:
138:
3054:
2566:
is ambiguous, as this can also mean the complex natural logarithmic
449:
Positive numbers less than 1 have negative logarithms. For example,
24:
2623:
2527:
1979:
1914:
1428:
3038:
1854:
to include only one entry for each mantissa. In the example of
2837:
2751:
2076:{\displaystyle \log _{10}(x)={\frac {\ln(x)}{\ln(10)}}\quad }
799:{\displaystyle \log _{10}(0.012)\approx 8.07918{\bmod {1}}0,}
781:
157:, an English mathematician who pioneered its use, as well as
2501:{\displaystyle {d \over dx}\log _{10}(x)={1 \over x\ln(10)}}
2307:
as procedures exist for determining the numerical value for
1427:* This step makes the mantissa between 0 and 1, so that its
2793:, the eccentricity of the earth ellipsoid (a small number).
2417:{\displaystyle {d \over dx}\log _{b}(x)={1 \over x\ln(b)}}
205:, it is printed as "log", but mathematicians usually mean
125:
A graph of the common logarithm of numbers from 0.1 to 100
3065:
Engineering
Acoustics: An Introduction to Noise Control
2587:
is generally used to describe the fractional part of a
2760:
2723:
2691:
2659:
2430:
2346:
2284:
2185:
2089:
2008:
1836:
1797:
1641:
1618:
821:
751:
716:
683:
611:
458:
319:
3086:
Handbook of mathematics for engineers and scientists
2591:
number on computers, though the recommended term is
1909:, the inventor of what are now called natural (base-
49:. Unsourced material may be challenged and removed.
3082:
2823:. Vol. Part I: Plane Trigonometry. New York:
2785:
2742:
2709:
2677:
2500:
2416:
2297:
2267:
2171:
2075:
1842:
1822:
1776:
1624:
1416:
798:
734:
698:
666:
586:
429:
3109:
3009:
1601:Note that the mantissa is common to all of the
2316:Natural logarithm § Efficient computation
293:
2887:
2750:, the minor radius of the earth ellipsoid in
1436:
286:as well. For the history of such tables, see
2856:Introductio in Analysin Infinitorum (Part 2)
1940:. Mathematicians, on the other hand, wrote "
2956:
2812:
2810:
2717:. From the context, it is understood that
2334:The derivative of a logarithm with a base
2324:Algorithms for computing binary logarithms
1270:
883:
882:
2893:
2831:
2816:
2622:
2285:
109:Learn how and when to remove this message
2807:
1978:
1864:
257:
120:
2962:
2278:using logarithms of any available base
1791:is a constant, the mantissa comes from
3110:
2985:"Derivatives of Logarithmic Functions"
2605:
444:
141:with base 10. It is also known as the
3061:
2937:
2966:Logarithmic and Trigonometric Tables
2933:
2931:
47:adding citations to reliable sources
18:
2786:{\displaystyle e=10^{8.9054355-10}}
13:
14:
3129:
2928:
2743:{\displaystyle b=10^{6.51335464}}
2678:{\displaystyle \log b=6.51335464}
2995:from the original on 2020-10-01.
2710:{\displaystyle \log e=8.9054355}
1974:
735:{\displaystyle {\bar {2}}.07918}
161:. Historically, it was known as
23:
3003:
2977:
2963:Hedrick, Earle Raymond (1913).
2653:gives (beginning of section 8)
2545:(also commonly called mantissa)
2264:
2186:
2168:
2090:
2072:
34:needs additional citations for
2904:
2845:; du Pasquier, Louis Gustave;
2598:
2573:
2556:
2492:
2486:
2465:
2459:
2408:
2402:
2381:
2375:
2258:
2252:
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2228:
2206:
2200:
2162:
2156:
2138:
2132:
2110:
2104:
2066:
2060:
2049:
2043:
2028:
2022:
1830:, which is constant for given
1817:
1811:
1762:
1756:
1703:
1697:
1609:. This holds for any positive
1404:
1398:
1374:
1362:
1341:
1335:
1229:
1217:
1211:
1199:
1188:
1173:
1167:
1152:
1135:
1117:
1103:
1097:
1078:
1072:
1051:
1039:
1013:
984:
978:
903:
897:
864:
852:
846:
771:
765:
723:
690:
643:
631:
625:
557:
551:
478:
472:
412:
406:
339:
333:
1:
2894:Scherffer, P. Carolo (1772).
2859:. 1 (in Latin). Vol. 9.
2800:
2329:
1823:{\displaystyle \log _{10}(x)}
306:The integer part, called the
2912:"Introduction to Logarithms"
7:
2511:
294:Mantissa and characteristic
149:, named after its base, or
10:
3134:
2754:(a large number), whereas
1890:
1886:
699:{\displaystyle {\bar {n}}}
3068:. Springer. p. 448.
2611:Astronomische Nachrichten
2849:; Trost, Ernst (1945) .
2641:10.1002/asna.18260041601
2549:
3062:Möser, Michael (2009).
1431:(10) can be looked up.
284:trigonometric functions
211:ISO 80000 specification
2825:Henry Holt and Company
2787:
2744:
2711:
2679:
2502:
2418:
2299:
2269:
2173:
2077:
1996:
1883:
1869:Numbers are placed on
1844:
1824:
1778:
1626:
1418:
800:
736:
700:
668:
588:
431:
268:mechanical calculators
263:
126:
2944:mathworld.wolfram.com
2788:
2745:
2712:
2680:
2579:This use of the word
2568:multi-valued function
2503:
2419:
2300:
2298:{\displaystyle \,B~.}
2270:
2174:
2078:
1982:
1893:History of logarithms
1868:
1845:
1825:
1779:
1627:
1419:
801:
737:
701:
669:
589:
432:
261:
169:. It is indicated by
163:logarithmus decimalis
124:
16:Mathematical function
2758:
2721:
2689:
2657:
2428:
2344:
2282:
2183:
2087:
2006:
1983:The logarithm keys (
1834:
1795:
1639:
1616:
819:
749:
714:
681:
609:
456:
317:
43:improve this article
2938:Weisstein, Eric W.
2633:1825AN......4..241B
2538:Napierian logarithm
1917:of his logarithms.
1852:table of logarithms
1439:
445:Negative logarithms
167:logarithmus decadis
151:Briggsian logarithm
3011:Abramowitz, Milton
2940:"Common Logarithm"
2916:www.mathsisfun.com
2783:
2740:
2707:
2675:
2498:
2414:
2295:
2265:
2169:
2073:
1997:
1948:" when they meant
1928:" when they meant
1884:
1840:
1820:
1774:
1622:
1437:
1414:
1412:
796:
732:
696:
664:
584:
427:
264:
225:should be written
159:standard logarithm
127:
58:"Common logarithm"
3100:978-1-58488-502-3
3075:978-3-540-92722-8
3030:978-0-486-61272-0
3015:Stegun, Irene Ann
2969:. New York, USA:
2533:Logarithmic scale
2496:
2444:
2412:
2360:
2291:
2262:
2166:
2070:
1843:{\displaystyle x}
1625:{\displaystyle x}
1611:real number
1599:
1598:
1138:
1120:
1016:
880:
867:
829:
726:
693:
646:
207:natural logarithm
147:decimal logarithm
143:decadic logarithm
119:
118:
111:
93:
3125:
3104:
3079:
3058:
3017:, eds. (1983) .
2997:
2996:
2981:
2975:
2974:
2960:
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2953:
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2950:
2935:
2926:
2925:
2923:
2922:
2908:
2902:
2901:
2891:
2885:
2884:
2878:
2874:
2872:
2864:
2851:Speiser, Andreas
2847:Brandt, Heinrich
2843:Speiser, Andreas
2835:
2829:
2828:
2814:
2794:
2792:
2790:
2789:
2784:
2782:
2781:
2749:
2747:
2746:
2741:
2739:
2738:
2716:
2714:
2713:
2708:
2684:
2682:
2681:
2676:
2652:
2626:
2602:
2596:
2577:
2571:
2565:
2560:
2518:Binary logarithm
2507:
2505:
2504:
2499:
2497:
2495:
2472:
2455:
2454:
2445:
2443:
2432:
2423:
2421:
2420:
2415:
2413:
2411:
2388:
2371:
2370:
2361:
2359:
2348:
2320:logarithm base 2
2312:
2304:
2302:
2301:
2296:
2289:
2274:
2272:
2271:
2266:
2263:
2261:
2248:
2247:
2237:
2224:
2223:
2213:
2196:
2195:
2178:
2176:
2175:
2170:
2167:
2165:
2152:
2151:
2141:
2128:
2127:
2117:
2100:
2099:
2082:
2080:
2079:
2074:
2071:
2069:
2052:
2035:
2018:
2017:
1994:
1990:
1987:for base-10 and
1970:
1961:
1947:
1939:
1927:
1881:
1879:
1861:
1859:
1850:. This allows a
1849:
1847:
1846:
1841:
1829:
1827:
1826:
1821:
1807:
1806:
1790:
1783:
1781:
1780:
1775:
1752:
1751:
1739:
1735:
1734:
1718:
1717:
1693:
1692:
1680:
1676:
1675:
1674:
1651:
1650:
1631:
1629:
1628:
1623:
1608:
1606:
1594:
1574:
1440:
1423:
1421:
1420:
1415:
1413:
1394:
1393:
1380:
1358:
1357:
1331:
1330:
1318:
1314:
1313:
1294:
1293:
1280:
1276:
1275:
1235:
1147:
1140:
1139:
1131:
1122:
1121:
1113:
1093:
1092:
1068:
1067:
1035:
1034:
1018:
1017:
1009:
974:
973:
952:
948:
941:
940:
920:
919:
893:
892:
881:
878:
869:
868:
860:
842:
841:
830:
827:
805:
803:
802:
797:
789:
788:
761:
760:
741:
739:
738:
733:
728:
727:
719:
709:
706:is read as "bar
705:
703:
702:
697:
695:
694:
686:
673:
671:
670:
665:
648:
647:
639:
621:
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593:
591:
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585:
547:
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525:
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493:
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468:
467:
436:
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428:
402:
401:
383:
379:
372:
371:
354:
353:
329:
328:
254:
246:
232:
224:
213:recommends that
200:
196:
188:
176:
135:common logarithm
114:
107:
103:
100:
94:
92:
51:
27:
19:
3133:
3132:
3128:
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3126:
3124:
3123:
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3108:
3107:
3101:
3076:
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3006:
3001:
3000:
2983:
2982:
2978:
2961:
2957:
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2936:
2929:
2920:
2918:
2910:
2909:
2905:
2892:
2888:
2876:
2875:
2866:
2865:
2839:Euler, Leonhard
2836:
2832:
2815:
2808:
2803:
2798:
2797:
2771:
2767:
2759:
2756:
2755:
2734:
2730:
2722:
2719:
2718:
2690:
2687:
2686:
2658:
2655:
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2603:
2599:
2578:
2574:
2563:
2561:
2557:
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2514:
2476:
2471:
2450:
2446:
2436:
2431:
2429:
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2425:
2392:
2387:
2366:
2362:
2352:
2347:
2345:
2342:
2341:
2332:
2310:
2309:logarithm base
2283:
2280:
2279:
2243:
2239:
2238:
2219:
2215:
2214:
2212:
2191:
2187:
2184:
2181:
2180:
2147:
2143:
2142:
2123:
2119:
2118:
2116:
2095:
2091:
2088:
2085:
2084:
2053:
2036:
2034:
2013:
2009:
2007:
2004:
2003:
1992:
1988:
1977:
1964:
1955:
1949:
1941:
1933:
1929:
1921:
1895:
1889:
1877:
1875:
1857:
1855:
1835:
1832:
1831:
1802:
1798:
1796:
1793:
1792:
1788:
1747:
1743:
1730:
1726:
1722:
1713:
1709:
1688:
1684:
1670:
1666:
1659:
1655:
1646:
1642:
1640:
1637:
1636:
1617:
1614:
1613:
1604:
1602:
1592:
1572:
1494:
1483:
1469:
1449:Characteristic
1411:
1410:
1389:
1385:
1378:
1377:
1353:
1349:
1344:
1326:
1322:
1306:
1302:
1298:
1289:
1285:
1278:
1277:
1271:
1269:
1252:
1233:
1232:
1191:
1145:
1144:
1130:
1129:
1112:
1111:
1106:
1088:
1084:
1063:
1059:
1054:
1030:
1026:
1023:
1022:
1008:
1007:
987:
969:
965:
933:
929:
928:
924:
915:
911:
906:
888:
884:
877:
874:
873:
859:
858:
837:
833:
831:
828:As found above,
826:
822:
820:
817:
816:
784:
780:
756:
752:
750:
747:
746:
718:
717:
715:
712:
711:
707:
685:
684:
682:
679:
678:
638:
637:
616:
612:
610:
607:
606:
542:
538:
506:
502:
501:
497:
488:
484:
463:
459:
457:
454:
453:
447:
397:
393:
367:
363:
362:
358:
349:
345:
324:
320:
318:
315:
314:
296:
248:
240:
234:
226:
218:
214:
198:
197:with a capital
190:
189:, or sometimes
182:
178:
170:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3131:
3121:
3120:
3106:
3105:
3099:
3080:
3074:
3059:
3029:
3005:
3002:
2999:
2998:
2991:. 2021-04-14.
2976:
2955:
2927:
2903:
2886:
2830:
2805:
2804:
2802:
2799:
2796:
2795:
2780:
2777:
2774:
2770:
2766:
2763:
2737:
2733:
2729:
2726:
2706:
2703:
2700:
2697:
2694:
2674:
2671:
2668:
2665:
2662:
2617:(8): 852–861.
2597:
2589:floating-point
2572:
2554:
2553:
2551:
2548:
2547:
2546:
2540:
2535:
2530:
2525:
2520:
2513:
2510:
2494:
2491:
2488:
2485:
2482:
2479:
2475:
2470:
2467:
2464:
2461:
2458:
2453:
2449:
2442:
2439:
2435:
2410:
2407:
2404:
2401:
2398:
2395:
2391:
2386:
2383:
2380:
2377:
2374:
2369:
2365:
2358:
2355:
2351:
2338:is such that
2331:
2328:
2294:
2288:
2276:
2275:
2260:
2257:
2254:
2251:
2246:
2242:
2236:
2233:
2230:
2227:
2222:
2218:
2211:
2208:
2205:
2202:
2199:
2194:
2190:
2164:
2161:
2158:
2155:
2150:
2146:
2140:
2137:
2134:
2131:
2126:
2122:
2115:
2112:
2109:
2106:
2103:
2098:
2094:
2068:
2065:
2062:
2059:
2056:
2051:
2048:
2045:
2042:
2039:
2033:
2030:
2027:
2024:
2021:
2016:
2012:
1976:
1973:
1951:
1931:
1891:Main article:
1888:
1885:
1839:
1819:
1816:
1813:
1810:
1805:
1801:
1785:
1784:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1750:
1746:
1742:
1738:
1733:
1729:
1725:
1721:
1716:
1712:
1708:
1705:
1702:
1699:
1696:
1691:
1687:
1683:
1679:
1673:
1669:
1665:
1662:
1658:
1654:
1649:
1645:
1621:
1597:
1596:
1590:
1587:
1584:
1583:−5.301 029...
1581:
1577:
1576:
1570:
1567:
1564:
1563:−0.301 029...
1561:
1557:
1556:
1553:
1550:
1547:
1544:
1540:
1539:
1536:
1533:
1530:
1527:
1523:
1522:
1519:
1516:
1513:
1510:
1506:
1505:
1503:
1492:
1489:
1481:
1475:
1467:
1464:
1457:
1456:
1455:Combined form
1453:
1450:
1447:
1444:
1425:
1424:
1409:
1406:
1403:
1400:
1397:
1392:
1388:
1384:
1381:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1356:
1352:
1348:
1345:
1343:
1340:
1337:
1334:
1329:
1325:
1321:
1317:
1312:
1309:
1305:
1301:
1297:
1292:
1288:
1284:
1281:
1279:
1274:
1268:
1265:
1262:
1259:
1256:
1253:
1251:
1248:
1245:
1242:
1239:
1236:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1146:
1143:
1137:
1134:
1128:
1125:
1119:
1116:
1110:
1107:
1105:
1102:
1099:
1096:
1091:
1087:
1083:
1080:
1077:
1074:
1071:
1066:
1062:
1058:
1055:
1053:
1050:
1047:
1044:
1041:
1038:
1033:
1029:
1025:
1024:
1021:
1015:
1012:
1006:
1003:
1000:
997:
994:
991:
988:
986:
983:
980:
977:
972:
968:
964:
961:
958:
955:
951:
947:
944:
939:
936:
932:
927:
923:
918:
914:
910:
907:
905:
902:
899:
896:
891:
887:
876:
875:
872:
866:
863:
857:
854:
851:
848:
845:
840:
836:
832:
825:
824:
807:
806:
795:
792:
787:
783:
779:
776:
773:
770:
767:
764:
759:
755:
731:
725:
722:
692:
689:
675:
674:
663:
660:
657:
654:
651:
645:
642:
636:
633:
630:
627:
624:
619:
615:
595:
594:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
545:
541:
537:
534:
531:
528:
524:
520:
517:
512:
509:
505:
500:
496:
491:
487:
483:
480:
477:
474:
471:
466:
462:
446:
443:
438:
437:
426:
423:
420:
417:
414:
411:
408:
405:
400:
396:
392:
389:
386:
382:
378:
375:
370:
366:
361:
357:
352:
348:
344:
341:
338:
335:
332:
327:
323:
308:characteristic
295:
292:
236:
216:
180:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3130:
3119:
3116:
3115:
3113:
3102:
3096:
3093:. p. 9.
3092:
3088:
3087:
3081:
3077:
3071:
3067:
3066:
3060:
3056:
3052:
3048:
3044:
3040:
3036:
3032:
3026:
3022:
3021:
3016:
3012:
3008:
3007:
2994:
2990:
2986:
2980:
2972:
2968:
2967:
2959:
2945:
2941:
2934:
2932:
2917:
2913:
2907:
2899:
2898:
2890:
2882:
2870:
2862:
2858:
2857:
2852:
2848:
2844:
2840:
2834:
2827:. p. 31.
2826:
2822:
2821:
2813:
2811:
2806:
2778:
2775:
2772:
2768:
2764:
2761:
2753:
2735:
2731:
2727:
2724:
2704:
2701:
2698:
2695:
2692:
2672:
2669:
2666:
2663:
2660:
2650:
2646:
2642:
2638:
2634:
2630:
2625:
2620:
2616:
2612:
2608:
2607:Bessel, F. W.
2604:For example,
2601:
2594:
2590:
2586:
2582:
2576:
2569:
2562:The notation
2559:
2555:
2544:
2541:
2539:
2536:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2515:
2509:
2489:
2483:
2480:
2477:
2473:
2468:
2462:
2456:
2451:
2447:
2440:
2437:
2433:
2405:
2399:
2396:
2393:
2389:
2384:
2378:
2372:
2367:
2363:
2356:
2353:
2349:
2339:
2337:
2327:
2325:
2321:
2317:
2313:
2305:
2292:
2286:
2255:
2249:
2244:
2240:
2231:
2225:
2220:
2216:
2209:
2203:
2197:
2192:
2188:
2159:
2153:
2148:
2144:
2135:
2129:
2124:
2120:
2113:
2107:
2101:
2096:
2092:
2063:
2057:
2054:
2046:
2040:
2037:
2031:
2025:
2019:
2014:
2010:
2002:
2001:
2000:
1986:
1981:
1975:Numeric value
1972:
1968:
1959:
1954:
1945:
1937:
1925:
1918:
1916:
1912:
1908:
1904:
1900:
1894:
1872:
1867:
1863:
1853:
1837:
1814:
1808:
1803:
1799:
1771:
1768:
1765:
1759:
1753:
1748:
1744:
1740:
1736:
1731:
1727:
1723:
1719:
1714:
1710:
1706:
1700:
1694:
1689:
1685:
1681:
1677:
1671:
1667:
1663:
1660:
1656:
1652:
1647:
1643:
1635:
1634:
1633:
1619:
1612:
1591:
1589:0.698 970...
1588:
1585:
1582:
1579:
1578:
1571:
1569:0.698 970...
1568:
1565:
1562:
1559:
1558:
1555:0.698 970...
1554:
1552:0.698 970...
1551:
1548:
1546:0.698 970...
1545:
1542:
1541:
1538:1.698 970...
1537:
1535:0.698 970...
1534:
1531:
1529:1.698 970...
1528:
1525:
1524:
1521:6.698 970...
1520:
1518:0.698 970...
1517:
1514:
1512:6.698 970...
1511:
1508:
1507:
1504:
1502:
1498:
1490:
1487:
1479:
1476:
1473:
1465:
1462:
1459:
1458:
1454:
1451:
1448:
1445:
1442:
1441:
1435:
1432:
1430:
1407:
1401:
1395:
1390:
1386:
1382:
1371:
1368:
1365:
1359:
1354:
1350:
1346:
1338:
1332:
1327:
1323:
1319:
1315:
1310:
1307:
1303:
1299:
1295:
1290:
1286:
1282:
1272:
1266:
1263:
1260:
1257:
1254:
1249:
1246:
1243:
1240:
1237:
1226:
1223:
1220:
1214:
1208:
1205:
1202:
1196:
1193:
1185:
1182:
1179:
1176:
1170:
1164:
1161:
1158:
1155:
1149:
1141:
1132:
1126:
1123:
1114:
1108:
1100:
1094:
1089:
1085:
1081:
1075:
1069:
1064:
1060:
1056:
1048:
1045:
1042:
1036:
1031:
1027:
1019:
1010:
1004:
1001:
998:
995:
992:
989:
981:
975:
970:
966:
962:
959:
956:
953:
949:
945:
942:
937:
934:
930:
925:
921:
916:
912:
908:
900:
894:
889:
885:
870:
861:
855:
849:
843:
838:
834:
815:
814:
813:
810:
793:
790:
785:
777:
774:
768:
762:
757:
753:
745:
744:
743:
729:
720:
687:
661:
658:
655:
652:
649:
640:
634:
628:
622:
617:
613:
605:
604:
603:
601:
600:bar notation,
581:
578:
575:
572:
569:
566:
563:
560:
554:
548:
543:
539:
535:
532:
529:
526:
522:
518:
515:
510:
507:
503:
498:
494:
489:
485:
481:
475:
469:
464:
460:
452:
451:
450:
442:
424:
421:
418:
415:
409:
403:
398:
394:
390:
387:
384:
380:
376:
373:
368:
364:
359:
355:
350:
346:
342:
336:
330:
325:
321:
313:
312:
311:
309:
304:
302:
291:
289:
285:
281:
277:
273:
269:
260:
256:
252:
244:
239:
230:
222:
212:
208:
204:
194:
186:
174:
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3085:
3064:
3019:
3004:Bibliography
2988:
2979:
2965:
2958:
2947:. Retrieved
2943:
2919:. Retrieved
2915:
2906:
2896:
2889:
2861:B.G. Teubner
2855:
2833:
2820:Trigonometry
2819:
2614:
2610:
2600:
2584:
2580:
2575:
2558:
2340:
2335:
2333:
2306:
2277:
1998:
1984:
1966:
1957:
1952:
1943:
1935:
1923:
1919:
1910:
1899:Henry Briggs
1896:
1786:
1600:
1595:.698 970...
1575:.698 970...
1500:
1496:
1485:
1477:
1471:
1460:
1433:
1426:
811:
808:
676:
599:
596:
448:
439:
307:
305:
300:
297:
265:
250:
242:
237:
228:
220:
192:
184:
172:
166:
162:
158:
155:Henry Briggs
150:
146:
142:
134:
128:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2877:|work=
2593:significand
2543:Significand
2523:Cologarithm
1903:John Napier
1480:= floor(log
710:", so that
203:calculators
145:and as the
131:mathematics
99:August 2020
3118:Logarithms
2949:2020-08-29
2921:2020-08-29
2801:References
2736:6.51335464
2673:6.51335464
2330:Derivative
1871:slide rule
1580:0.000 005
1509:5 000 000
1446:Logarithm
276:slide rule
247:should be
69:newspapers
3091:CRC Press
2971:Macmillan
2879:ignored (
2869:cite book
2776:−
2773:8.9054355
2705:8.9054355
2696:
2664:
2649:118630614
2624:0908.1823
2484:
2457:
2400:
2373:
2250:
2226:
2198:
2154:
2130:
2102:
2058:
2041:
2020:
1991:for base-
1907:Edinburgh
1809:
1754:
1720:
1695:
1664:×
1653:
1463:= 5 × 10
1452:Mantissa
1396:
1369:×
1360:
1333:
1308:−
1296:
1283:≈
1273:∗
1258:−
1241:−
1197:−
1177:−
1156:−
1136:¯
1118:¯
1109:≈
1095:
1070:
1046:×
1037:
1014:¯
993:−
990:≈
976:
957:−
943:×
935:−
922:
895:
865:¯
856:≈
844:
775:≈
763:
724:¯
691:¯
659:−
644:¯
635:≈
623:
602:is used:
579:−
564:−
561:≈
549:
530:−
516:×
508:−
495:
470:
416:≈
404:
374:×
356:
331:
288:log table
139:logarithm
3112:Category
3055:65-12253
3039:64-60036
2993:Archived
2585:mantissa
2581:mantissa
2512:See also
1632:because
662:1.92082.
582:1.92082.
425:0.07918.
301:mantissa
153:, after
3047:0167642
2853:(ed.).
2629:Bibcode
2528:Decibel
1915:chiliad
1887:History
1443:Number
1429:antilog
1267:0.00860
1250:1.00860
1227:0.92942
1221:0.07918
1186:0.92942
1165:0.07918
1002:0.92942
778:8.07918
653:0.07918
573:0.07918
137:is the
83:scholar
3097:
3072:
3053:
3045:
3037:
3027:
2989:Math24
2647:
2318:) and
2290:
1787:Since
1402:0.0102
1142:.92942
1124:.07918
1020:.92942
871:.07918
730:.07918
280:tables
272:tables
233:, and
133:, the
85:
78:
71:
64:
56:
2752:toise
2645:S2CID
2619:arXiv
2550:Notes
2424:, so
2322:(see
2314:(see
1880:3 = 6
1076:0.012
1043:0.012
879:Since
850:0.012
769:0.012
629:0.012
476:0.012
201:; on
90:JSTOR
76:books
3095:ISBN
3070:ISBN
3051:LCCN
3035:LCCN
3025:ISBN
2881:help
2179:or
2083:or
1942:log(
1922:log(
1560:0.5
1499:) −
1372:1.02
1366:0.01
1339:1.02
1101:0.85
1049:0.85
901:0.85
191:Log(
171:log(
62:news
2693:log
2661:log
2637:doi
2615:331
2564:Log
2448:log
2364:log
2326:).
2241:log
2217:log
2189:log
2145:log
2121:log
2093:log
2011:log
1985:log
1965:ln(
1950:log
1930:log
1905:at
1800:log
1745:log
1711:log
1686:log
1644:log
1586:−6
1566:−1
1526:50
1491:log
1488:))
1466:log
1387:log
1351:log
1324:log
1287:log
1086:log
1061:log
1028:log
982:8.5
967:log
946:8.5
913:log
886:log
835:log
782:mod
754:log
614:log
555:1.2
540:log
519:1.2
486:log
461:log
410:1.2
395:log
377:1.2
347:log
337:120
322:log
249:ln(
235:log
227:lg(
215:log
179:log
165:or
129:In
45:by
3114::
3089:.
3049:.
3043:MR
3041:.
3033:.
3013:;
2987:.
2942:.
2930:^
2914:.
2873::
2871:}}
2867:{{
2841:;
2809:^
2779:10
2769:10
2732:10
2685:,
2643:.
2635:.
2627:.
2613:.
2508:.
2490:10
2481:ln
2452:10
2397:ln
2256:10
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