46:
1245:
2820:
states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integer powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of an
2710:
2052:
3184:
2404:
1810:
1912:
1073:
2537:
2320:
2240:
3293:
1009:
1670:
810:
1078:
439:
241:
133:
2913:
2796:
2133:
1447:
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the
3031:
1411:
632:
381:
1349:
1285:, but 2 to the 4 is 8 (or 4,096) whereas 2 to the 3 is 2 (or 2,417,851,639,229,258,349,412,352). When no parentheses are written, by convention the order is top-down, not bottom-up:
705:
2982:
1559:
1510:
2563:
869:
1918:
183:
921:
1047:
947:
551:
892:
834:
71:
510:
3047:
2326:
1677:
1821:
1463:. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions
1240:{\displaystyle {\begin{aligned}b^{m+n}&=b^{m}\cdot b^{n}\\(b^{m})^{n}&=b^{m\cdot n}\\(b\cdot c)^{n}&=b^{n}\cdot c^{n}\end{aligned}}}
740:
These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the
3750:
3729:
3710:
3540:
3389:
Pratt, Vaughan, "Algebra", The
Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL:
2445:
3805:
2251:
3677:
3606:
2140:
3800:
716:
3195:
3316:
952:
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity
955:
1583:
3767:
754:
3298:
So, these formulas are identities in every monoid. As for any equality, the formulas without quantifier are often called
386:
188:
3776:
76:
3795:
2854:
2739:
2064:
745:
2987:
2548:
1354:
560:
309:
2832:
gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve
1291:
3527:
Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability
637:
260:
2938:
2813:
1422:
722:
458:
303:
279:
256:
3378:
2705:{\displaystyle \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.}
3630:
2829:
2544:
1515:
1466:
2047:{\displaystyle \log _{2}(16)=\log _{2}\!\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4}
1049:), which can be used to break down expressions of larger angles into those with smaller constituents.
2845:
2822:
50:
842:
3311:
2848:
2807:
283:
142:
3657:
3577:
897:
45:
3614:
1014:
926:
3746:
3725:
3706:
3673:
3626:
3602:
3536:
3321:
515:
306:, and an identity is an equality between functions that are differently defined. For example,
35:
877:
819:
56:
3740:
3569:
3033:
is often left implicit, when it is stated that the formula is an identity. For example, the
31:
3302:. In other words, an identity is an equation that is true for all values of the variables.
483:
3780:
3653:
3634:
3179:{\displaystyle \forall x,y,z:x*(y*z)=(x*y)*z,\quad \forall x:x*1=x,\quad \forall x:1*x=x,}
2399:{\displaystyle \log _{10}\!{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}
39:
2833:
1058:
17:
3789:
3525:
3474:
2932:
2812:
The hyperbolic functions satisfy many identities, all of them similar in form to the
751:
One of the most prominent examples of trigonometric identities involves the equation
475:
3641:. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 243–320.
3450:
1805:{\displaystyle \log _{3}(243)=\log _{3}(9\cdot 27)=\log _{3}(9)+\log _{3}(27)=2+3=5}
3661:
3390:
2928:
730:
744:
of non-trigonometric functions: a common technique which involves first using the
3700:
3596:
3532:
3499:
2557:
can be determined using either of these two logarithms by the previous formula:
1907:{\displaystyle \log _{b}\!\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}
1274:
1251:
813:
454:
248:
136:
3773:
442:
748:, and then simplifying the resulting integral with a trigonometric identity.
3672:
Monographs on
Theoretical Computer Science. Vol. 25. Berlin: Springer.
3426:
1436:
3402:
282:) produce the same value for all values of the variables within a certain
3299:
741:
734:
707:, can be useful in simplifying algebraic expressions and expanding them.
3581:
1064:
554:
3379:
http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
3038:
3573:
3557:
733:, which are identities involving both angles and side lengths of a
3669:
3034:
2532:{\displaystyle \log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.}
726:
44:
2315:{\displaystyle \log _{b}\!{\sqrt{x}}={\frac {\log _{b}(x)}{p}}}
1273:
Also unlike addition and multiplication, exponentiation is not
2235:{\displaystyle \log _{2}(64)=\log _{2}(2^{6})=6\log _{2}(2)=6}
1455:
times the logarithm of the number itself; the logarithm of a
3288:{\displaystyle x*(y*z)=(x*y)*z,\qquad x*1=x,\qquad 1*x=x.}
1250:
Unlike addition and multiplication, exponentiation is not
725:
are identities involving certain functions of one or more
441:
are identities. Identities are sometimes indicated by the
27:
Equation that is satisfied for all values of the variables
3770:
Online encyclopedia of mathematical identities (archived)
1004:{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta }
3427:"Identity – math word definition – Math Open Reference"
1665:{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}
3198:
3050:
2990:
2941:
2857:
2742:
2566:
2448:
2329:
2254:
2143:
2067:
1921:
1824:
1680:
1586:
1518:
1469:
1357:
1294:
1076:
1017:
958:
929:
900:
880:
845:
822:
757:
640:
563:
518:
486:
389:
312:
191:
145:
79:
59:
805:{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}
894:, not all. For example, this equation is true when
434:{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}
236:{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}
3524:
3287:
3178:
3025:
2976:
2907:
2790:
2704:
2531:
2398:
2314:
2234:
2127:
2046:
1906:
1804:
1664:
1553:
1504:
1459:th root is the logarithm of the number divided by
1405:
1343:
1239:
1041:
1003:
941:
915:
886:
863:
828:
804:
699:
626:
545:
504:
433:
375:
235:
177:
127:
65:
2340:
2265:
1957:
1835:
128:{\displaystyle (x,y)=(\cos \theta ,\sin \theta )}
3601:(3rd ed.). Cengage Learning. p. 1155.
3391:https://plato.stanford.edu/entries/algebra/#Laws
2908:{\displaystyle \forall x_{1},\ldots ,x_{n}:s=t,}
3345:All statements in this section can be found in
1067:exponents, provided that the base is non-zero:
746:substitution rule with a trigonometric function
737:. Only the former are covered in this article.
2791:{\displaystyle b=x^{\frac {1}{\log _{b}(x)}}.}
2128:{\displaystyle \log _{b}(x^{p})=p\log _{b}(x)}
266: to another mathematical expression
1427:Several important formulas, sometimes called
8:
3523:Bernstein, Stephen; Bernstein, Ruth (1999),
3377:Equation. Encyclopedia of Mathematics. URL:
3354:
3026:{\displaystyle \forall x_{1},\ldots ,x_{n}}
476:Factorization § Recognizable patterns
3666:Universal Algebra for Computer Scientists
3197:
3049:
3017:
2998:
2989:
2965:
2946:
2940:
2884:
2865:
2856:
2763:
2753:
2741:
2678:
2654:
2647:
2623:
2599:
2592:
2571:
2565:
2547:calculate the logarithms to bases 10 and
2505:
2481:
2474:
2453:
2447:
2426:) can be computed from the logarithms of
2380:
2365:
2351:
2341:
2334:
2328:
2288:
2281:
2271:
2266:
2259:
2253:
2208:
2189:
2173:
2148:
2142:
2107:
2088:
2072:
2066:
2008:
1983:
1962:
1951:
1926:
1920:
1886:
1861:
1840:
1829:
1823:
1766:
1741:
1710:
1685:
1679:
1644:
1619:
1591:
1585:
1534:
1529:
1517:
1485:
1480:
1468:
1406:{\displaystyle (b^{p})^{q}=b^{p\cdot q}.}
1388:
1375:
1365:
1356:
1327:
1319:
1304:
1299:
1293:
1227:
1214:
1197:
1165:
1148:
1138:
1121:
1108:
1085:
1077:
1075:
1016:
957:
928:
899:
879:
844:
821:
781:
762:
756:
658:
645:
639:
627:{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}
618:
593:
580:
562:
517:
485:
413:
394:
388:
376:{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}
367:
342:
329:
311:
215:
196:
190:
163:
150:
144:
78:
58:
1563:
3558:"109. Mnemonic for Hyperbolic Formulae"
3370:
3350:
3346:
3338:
1344:{\displaystyle b^{p^{q}}:=b^{(p^{q})},}
2553:. Logarithms with respect to any base
1063:The following identities hold for all
700:{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
3531:, Schaum's outline series, New York:
7:
3774:A Collection of Algebraic Identities
2977:{\displaystyle x_{1},\ldots ,x_{n}.}
3598:Technical mathematics with calculus
874:is only true for certain values of
3720:Kate, S.K.; Bhapkar, H.R. (2009).
3149:
3121:
3051:
2991:
2858:
2434:with respect to an arbitrary base
836:. On the other hand, the equation
557:, while other identities, such as
25:
3684:Here: Def.1 of Sect.3.2.1, p.160.
1554:{\displaystyle y=b^{\log _{b}y},}
1505:{\displaystyle x=b^{\log _{b}x},}
1443:Product, quotient, power and root
3041:are often given as the formulas
2844:Formally, an identity is a true
717:List of trigonometric identities
3595:Peterson, John Charles (2003).
3317:List of mathematical identities
3266:
3247:
3148:
3120:
139:, which satisfies the equation
3705:. Barrons Educational Series.
3633:(1990). "Rewrite Systems". In
3235:
3223:
3217:
3205:
3108:
3096:
3090:
3078:
2802:Hyperbolic function identities
2778:
2772:
2693:
2687:
2669:
2663:
2638:
2632:
2614:
2608:
2586:
2580:
2520:
2514:
2496:
2490:
2468:
2462:
2303:
2297:
2223:
2217:
2195:
2182:
2163:
2157:
2122:
2116:
2094:
2081:
2023:
2017:
1998:
1992:
1941:
1935:
1901:
1895:
1876:
1870:
1781:
1775:
1756:
1750:
1731:
1719:
1700:
1694:
1659:
1653:
1634:
1628:
1609:
1600:
1372:
1358:
1333:
1320:
1283:(2 · 3) · 4 = 2 · (3 · 4) = 24
1194:
1181:
1145:
1131:
1036:
1024:
974:
965:
864:{\displaystyle \cos \theta =1}
694:
682:
679:
667:
577:
564:
534:
525:
326:
313:
122:
98:
92:
80:
1:
3742:Adventures in Problem Solving
3556:Osborn, G. (1 January 1902).
2438:using the following formula:
1279:(2 + 3) + 4 = 2 + (3 + 4) = 9
457:. Formally, an identity is a
178:{\displaystyle x^{2}+y^{2}=1}
3768:The Encyclopedia of Equation
2825:number of hyperbolic sines.
480:Certain identities, such as
3639:Formal Models and Semantics
3357:, p. 1-1, for example.
2840:Logic and universal algebra
1011:, the addition formula for
3822:
3724:. Technical Publications.
3652:Wolfgang Wechsler (1992).
3500:"Trigonometric Identities"
2805:
1420:
1056:
916:{\displaystyle \theta =0,}
714:
473:
278:(which might contain some
29:
3806:Equivalence (mathematics)
3699:Downing, Douglas (2003).
1042:{\displaystyle \tan(x+y)}
942:{\displaystyle \theta =2}
729:. They are distinct from
3562:The Mathematical Gazette
2814:trigonometric identities
2733:, the base is given by:
1561:in the left hand sides.
1451:th power of a number is
723:trigonometric identities
711:Trigonometric identities
546:{\displaystyle a+(-a)=0}
30:Not to be confused with
3801:Mathematical identities
3355:Kate & Bhapkar 2009
887:{\displaystyle \theta }
829:{\displaystyle \theta }
261:mathematical expression
66:{\displaystyle \theta }
18:Mathematical identities
3745:. Universities Press.
3475:"Algebraic Identities"
3289:
3180:
3027:
2984:The quantifier prefix
2978:
2909:
2846:universally quantified
2792:
2706:
2545:scientific calculators
2533:
2400:
2316:
2236:
2129:
2048:
1908:
1806:
1666:
1555:
1506:
1429:logarithmic identities
1423:Logarithmic identities
1417:Logarithmic identities
1407:
1351: whereas
1345:
1241:
1053:Exponential identities
1043:
1005:
943:
917:
888:
865:
830:
812:which is true for all
806:
701:
628:
547:
506:
459:universally quantified
435:
377:
244:
237:
179:
129:
67:
3722:Basics Of Mathematics
3631:Jean-Pierre Jouannaud
3615:Chapter 26, page 1155
3403:"Mathwords: Identity"
3290:
3181:
3028:
2979:
2910:
2830:Gudermannian function
2793:
2729:) to an unknown base
2719:and its logarithm log
2707:
2534:
2401:
2317:
2237:
2130:
2049:
1909:
1807:
1667:
1556:
1507:
1408:
1346:
1277:either. For example,
1242:
1044:
1006:
944:
918:
889:
866:
831:
807:
702:
629:
548:
507:
505:{\displaystyle a+0=a}
436:
378:
238:
180:
130:
68:
48:
3739:Shirali, S. (2002).
3702:Algebra the Easy Way
3196:
3048:
2988:
2939:
2855:
2740:
2564:
2446:
2327:
2252:
2141:
2065:
1919:
1822:
1678:
1584:
1516:
1467:
1355:
1292:
1074:
1015:
956:
927:
898:
878:
843:
820:
755:
638:
561:
553:, form the basis of
516:
484:
470:Algebraic identities
387:
310:
189:
143:
77:
57:
51:Pythagorean identity
49:Visual proof of the
3498:Stapel, Elizabeth.
3431:www.mathopenref.com
3312:Accounting identity
2808:Hyperbolic function
731:triangle identities
284:domain of discourse
3796:Elementary algebra
3779:2011-10-01 at the
3658:Grzegorz Rozenberg
3451:"Basic Identities"
3353:, p. 275, or
3285:
3176:
3023:
2974:
2905:
2788:
2702:
2529:
2396:
2312:
2232:
2125:
2044:
1904:
1802:
1662:
1551:
1502:
1403:
1341:
1237:
1235:
1039:
1001:
939:
913:
884:
861:
826:
802:
697:
624:
543:
502:
431:
373:
294:is an identity if
286:. In other words,
245:
233:
175:
125:
63:
3752:978-81-7371-413-9
3731:978-81-8431-755-8
3712:978-0-7641-1972-9
3627:Nachum Dershowitz
3542:978-0-07-005023-5
3407:www.mathwords.com
3322:Law (mathematics)
2782:
2697:
2642:
2524:
2416:The logarithm log
2409:
2408:
2388:
2359:
2346:
2310:
2276:
1970:
1848:
1260:2 · 3 = 3 · 2 = 6
1256:2 + 3 = 3 + 2 = 5
465:Common identities
36:identity function
16:(Redirected from
3813:
3756:
3735:
3716:
3685:
3683:
3649:
3643:
3642:
3623:
3617:
3612:
3592:
3586:
3585:
3553:
3547:
3545:
3530:
3520:
3514:
3513:
3511:
3510:
3495:
3489:
3488:
3486:
3485:
3471:
3465:
3464:
3462:
3461:
3447:
3441:
3440:
3438:
3437:
3423:
3417:
3416:
3414:
3413:
3399:
3393:
3387:
3381:
3375:
3358:
3343:
3294:
3292:
3291:
3286:
3185:
3183:
3182:
3177:
3032:
3030:
3029:
3024:
3022:
3021:
3003:
3002:
2983:
2981:
2980:
2975:
2970:
2969:
2951:
2950:
2926:
2920:
2914:
2912:
2911:
2906:
2889:
2888:
2870:
2869:
2797:
2795:
2794:
2789:
2784:
2783:
2781:
2768:
2767:
2754:
2711:
2709:
2708:
2703:
2698:
2696:
2683:
2682:
2672:
2659:
2658:
2648:
2643:
2641:
2628:
2627:
2617:
2604:
2603:
2593:
2576:
2575:
2539:
2538:
2536:
2535:
2530:
2525:
2523:
2510:
2509:
2499:
2486:
2485:
2475:
2458:
2457:
2405:
2403:
2402:
2397:
2389:
2381:
2370:
2369:
2360:
2352:
2347:
2342:
2339:
2338:
2321:
2319:
2318:
2313:
2311:
2306:
2293:
2292:
2282:
2277:
2275:
2267:
2264:
2263:
2241:
2239:
2238:
2233:
2213:
2212:
2194:
2193:
2178:
2177:
2153:
2152:
2135:
2134:
2132:
2131:
2126:
2112:
2111:
2093:
2092:
2077:
2076:
2053:
2051:
2050:
2045:
2013:
2012:
1988:
1987:
1975:
1971:
1963:
1956:
1955:
1931:
1930:
1913:
1911:
1910:
1905:
1891:
1890:
1866:
1865:
1853:
1849:
1841:
1834:
1833:
1811:
1809:
1808:
1803:
1771:
1770:
1746:
1745:
1715:
1714:
1690:
1689:
1672:
1671:
1669:
1668:
1663:
1649:
1648:
1624:
1623:
1596:
1595:
1564:
1560:
1558:
1557:
1552:
1547:
1546:
1539:
1538:
1511:
1509:
1508:
1503:
1498:
1497:
1490:
1489:
1462:
1458:
1454:
1450:
1439:to one another:
1412:
1410:
1409:
1404:
1399:
1398:
1380:
1379:
1370:
1369:
1350:
1348:
1347:
1342:
1337:
1336:
1332:
1331:
1311:
1310:
1309:
1308:
1284:
1280:
1269:
1265:
1261:
1257:
1246:
1244:
1243:
1238:
1236:
1232:
1231:
1219:
1218:
1202:
1201:
1176:
1175:
1153:
1152:
1143:
1142:
1126:
1125:
1113:
1112:
1096:
1095:
1048:
1046:
1045:
1040:
1010:
1008:
1007:
1002:
948:
946:
945:
940:
922:
920:
919:
914:
893:
891:
890:
885:
870:
868:
867:
862:
835:
833:
832:
827:
811:
809:
808:
803:
786:
785:
767:
766:
706:
704:
703:
698:
663:
662:
650:
649:
633:
631:
630:
625:
623:
622:
598:
597:
585:
584:
552:
550:
549:
544:
511:
509:
508:
503:
452:
448:
440:
438:
437:
432:
418:
417:
399:
398:
382:
380:
379:
374:
372:
371:
347:
346:
334:
333:
302:define the same
242:
240:
239:
234:
220:
219:
201:
200:
184:
182:
181:
176:
168:
167:
155:
154:
134:
132:
131:
126:
72:
70:
69:
64:
53:: for any angle
32:identity element
21:
3821:
3820:
3816:
3815:
3814:
3812:
3811:
3810:
3786:
3785:
3781:Wayback Machine
3764:
3759:
3753:
3738:
3732:
3719:
3713:
3698:
3694:
3689:
3688:
3680:
3654:Wilfried Brauer
3651:
3650:
3646:
3635:Jan van Leeuwen
3625:
3624:
3620:
3609:
3594:
3593:
3589:
3574:10.2307/3602492
3555:
3554:
3550:
3543:
3522:
3521:
3517:
3508:
3506:
3497:
3496:
3492:
3483:
3481:
3479:www.sosmath.com
3473:
3472:
3468:
3459:
3457:
3449:
3448:
3444:
3435:
3433:
3425:
3424:
3420:
3411:
3409:
3401:
3400:
3396:
3388:
3384:
3376:
3372:
3367:
3362:
3361:
3344:
3340:
3335:
3330:
3308:
3194:
3193:
3046:
3045:
3013:
2994:
2986:
2985:
2961:
2942:
2937:
2936:
2922:
2916:
2880:
2861:
2853:
2852:
2842:
2834:complex numbers
2810:
2804:
2759:
2758:
2749:
2738:
2737:
2724:
2715:Given a number
2674:
2673:
2650:
2649:
2619:
2618:
2595:
2594:
2567:
2562:
2561:
2501:
2500:
2477:
2476:
2449:
2444:
2443:
2442:
2421:
2414:
2361:
2330:
2325:
2324:
2284:
2283:
2255:
2250:
2249:
2204:
2185:
2169:
2144:
2139:
2138:
2103:
2084:
2068:
2063:
2062:
2061:
2004:
1979:
1958:
1947:
1922:
1917:
1916:
1882:
1857:
1836:
1825:
1820:
1819:
1762:
1737:
1706:
1681:
1676:
1675:
1640:
1615:
1587:
1582:
1581:
1580:
1530:
1525:
1514:
1513:
1481:
1476:
1465:
1464:
1460:
1456:
1452:
1448:
1445:
1425:
1419:
1384:
1371:
1361:
1353:
1352:
1323:
1315:
1300:
1295:
1290:
1289:
1282:
1278:
1267:
1263:
1259:
1255:
1254:. For example,
1234:
1233:
1223:
1210:
1203:
1193:
1178:
1177:
1161:
1154:
1144:
1134:
1128:
1127:
1117:
1104:
1097:
1081:
1072:
1071:
1061:
1055:
1013:
1012:
954:
953:
925:
924:
923:but false when
896:
895:
876:
875:
841:
840:
818:
817:
777:
758:
753:
752:
721:Geometrically,
719:
713:
654:
641:
636:
635:
614:
589:
576:
559:
558:
514:
513:
482:
481:
478:
472:
467:
450:
446:
409:
390:
385:
384:
363:
338:
325:
308:
307:
211:
192:
187:
186:
159:
146:
141:
140:
75:
74:
55:
54:
43:
40:identity matrix
28:
23:
22:
15:
12:
11:
5:
3819:
3817:
3809:
3808:
3803:
3798:
3788:
3787:
3784:
3783:
3771:
3763:
3762:External links
3760:
3758:
3757:
3751:
3736:
3730:
3717:
3711:
3695:
3693:
3690:
3687:
3686:
3678:
3644:
3618:
3607:
3587:
3548:
3541:
3515:
3490:
3466:
3442:
3418:
3394:
3382:
3369:
3368:
3366:
3363:
3360:
3359:
3337:
3336:
3334:
3331:
3329:
3326:
3325:
3324:
3319:
3314:
3307:
3304:
3296:
3295:
3284:
3281:
3278:
3275:
3272:
3269:
3265:
3262:
3259:
3256:
3253:
3250:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3187:
3186:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3020:
3016:
3012:
3009:
3006:
3001:
2997:
2993:
2973:
2968:
2964:
2960:
2957:
2954:
2949:
2945:
2933:free variables
2931:with no other
2904:
2901:
2898:
2895:
2892:
2887:
2883:
2879:
2876:
2873:
2868:
2864:
2860:
2841:
2838:
2806:Main article:
2803:
2800:
2799:
2798:
2787:
2780:
2777:
2774:
2771:
2766:
2762:
2757:
2752:
2748:
2745:
2720:
2713:
2712:
2701:
2695:
2692:
2689:
2686:
2681:
2677:
2671:
2668:
2665:
2662:
2657:
2653:
2646:
2640:
2637:
2634:
2631:
2626:
2622:
2616:
2613:
2610:
2607:
2602:
2598:
2591:
2588:
2585:
2582:
2579:
2574:
2570:
2541:
2540:
2528:
2522:
2519:
2516:
2513:
2508:
2504:
2498:
2495:
2492:
2489:
2484:
2480:
2473:
2470:
2467:
2464:
2461:
2456:
2452:
2417:
2413:
2412:Change of base
2410:
2407:
2406:
2395:
2392:
2387:
2384:
2379:
2376:
2373:
2368:
2364:
2358:
2355:
2350:
2345:
2337:
2333:
2322:
2309:
2305:
2302:
2299:
2296:
2291:
2287:
2280:
2274:
2270:
2262:
2258:
2247:
2243:
2242:
2231:
2228:
2225:
2222:
2219:
2216:
2211:
2207:
2203:
2200:
2197:
2192:
2188:
2184:
2181:
2176:
2172:
2168:
2165:
2162:
2159:
2156:
2151:
2147:
2136:
2124:
2121:
2118:
2115:
2110:
2106:
2102:
2099:
2096:
2091:
2087:
2083:
2080:
2075:
2071:
2059:
2055:
2054:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2011:
2007:
2003:
2000:
1997:
1994:
1991:
1986:
1982:
1978:
1974:
1969:
1966:
1961:
1954:
1950:
1946:
1943:
1940:
1937:
1934:
1929:
1925:
1914:
1903:
1900:
1897:
1894:
1889:
1885:
1881:
1878:
1875:
1872:
1869:
1864:
1860:
1856:
1852:
1847:
1844:
1839:
1832:
1828:
1817:
1813:
1812:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1769:
1765:
1761:
1758:
1755:
1752:
1749:
1744:
1740:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1713:
1709:
1705:
1702:
1699:
1696:
1693:
1688:
1684:
1673:
1661:
1658:
1655:
1652:
1647:
1643:
1639:
1636:
1633:
1630:
1627:
1622:
1618:
1614:
1611:
1608:
1605:
1602:
1599:
1594:
1590:
1578:
1574:
1573:
1570:
1567:
1550:
1545:
1542:
1537:
1533:
1528:
1524:
1521:
1501:
1496:
1493:
1488:
1484:
1479:
1475:
1472:
1444:
1441:
1421:Main article:
1418:
1415:
1414:
1413:
1402:
1397:
1394:
1391:
1387:
1383:
1378:
1374:
1368:
1364:
1360:
1340:
1335:
1330:
1326:
1322:
1318:
1314:
1307:
1303:
1298:
1248:
1247:
1230:
1226:
1222:
1217:
1213:
1209:
1206:
1204:
1200:
1196:
1192:
1189:
1186:
1183:
1180:
1179:
1174:
1171:
1168:
1164:
1160:
1157:
1155:
1151:
1147:
1141:
1137:
1133:
1130:
1129:
1124:
1120:
1116:
1111:
1107:
1103:
1100:
1098:
1094:
1091:
1088:
1084:
1080:
1079:
1059:Exponentiation
1057:Main article:
1054:
1051:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
938:
935:
932:
912:
909:
906:
903:
883:
872:
871:
860:
857:
854:
851:
848:
825:
801:
798:
795:
792:
789:
784:
780:
776:
773:
770:
765:
761:
715:Main article:
712:
709:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
661:
657:
653:
648:
644:
621:
617:
613:
610:
607:
604:
601:
596:
592:
588:
583:
579:
575:
572:
569:
566:
542:
539:
536:
533:
530:
527:
524:
521:
501:
498:
495:
492:
489:
471:
468:
466:
463:
430:
427:
424:
421:
416:
412:
408:
405:
402:
397:
393:
370:
366:
362:
359:
356:
353:
350:
345:
341:
337:
332:
328:
324:
321:
318:
315:
232:
229:
226:
223:
218:
214:
210:
207:
204:
199:
195:
174:
171:
166:
162:
158:
153:
149:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
91:
88:
85:
82:
62:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3818:
3807:
3804:
3802:
3799:
3797:
3794:
3793:
3791:
3782:
3778:
3775:
3772:
3769:
3766:
3765:
3761:
3754:
3748:
3744:
3743:
3737:
3733:
3727:
3723:
3718:
3714:
3708:
3704:
3703:
3697:
3696:
3691:
3681:
3679:3-540-54280-9
3675:
3671:
3667:
3663:
3659:
3655:
3648:
3645:
3640:
3636:
3632:
3628:
3622:
3619:
3616:
3610:
3608:0-7668-6189-9
3604:
3600:
3599:
3591:
3588:
3583:
3579:
3575:
3571:
3567:
3563:
3559:
3552:
3549:
3544:
3538:
3534:
3529:
3528:
3519:
3516:
3505:
3501:
3494:
3491:
3480:
3476:
3470:
3467:
3456:
3452:
3446:
3443:
3432:
3428:
3422:
3419:
3408:
3404:
3398:
3395:
3392:
3386:
3383:
3380:
3374:
3371:
3364:
3356:
3352:
3349:, Section 4,
3348:
3342:
3339:
3332:
3327:
3323:
3320:
3318:
3315:
3313:
3310:
3309:
3305:
3303:
3301:
3282:
3279:
3276:
3273:
3270:
3267:
3263:
3260:
3257:
3254:
3251:
3248:
3244:
3241:
3238:
3232:
3229:
3226:
3220:
3214:
3211:
3208:
3202:
3199:
3192:
3191:
3190:
3189:or, shortly,
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3117:
3114:
3111:
3105:
3102:
3099:
3093:
3087:
3084:
3081:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3044:
3043:
3042:
3040:
3036:
3018:
3014:
3010:
3007:
3004:
2999:
2995:
2971:
2966:
2962:
2958:
2955:
2952:
2947:
2943:
2934:
2930:
2925:
2919:
2902:
2899:
2896:
2893:
2890:
2885:
2881:
2877:
2874:
2871:
2866:
2862:
2850:
2847:
2839:
2837:
2835:
2831:
2826:
2824:
2819:
2818:Osborn's rule
2815:
2809:
2801:
2785:
2775:
2769:
2764:
2760:
2755:
2750:
2746:
2743:
2736:
2735:
2734:
2732:
2728:
2723:
2718:
2699:
2690:
2684:
2679:
2675:
2666:
2660:
2655:
2651:
2644:
2635:
2629:
2624:
2620:
2611:
2605:
2600:
2596:
2589:
2583:
2577:
2572:
2568:
2560:
2559:
2558:
2556:
2552:
2551:
2546:
2526:
2517:
2511:
2506:
2502:
2493:
2487:
2482:
2478:
2471:
2465:
2459:
2454:
2450:
2441:
2440:
2439:
2437:
2433:
2429:
2425:
2420:
2411:
2393:
2390:
2385:
2382:
2377:
2374:
2371:
2366:
2362:
2356:
2353:
2348:
2343:
2335:
2331:
2323:
2307:
2300:
2294:
2289:
2285:
2278:
2272:
2268:
2260:
2256:
2248:
2245:
2244:
2229:
2226:
2220:
2214:
2209:
2205:
2201:
2198:
2190:
2186:
2179:
2174:
2170:
2166:
2160:
2154:
2149:
2145:
2137:
2119:
2113:
2108:
2104:
2100:
2097:
2089:
2085:
2078:
2073:
2069:
2060:
2057:
2056:
2041:
2038:
2035:
2032:
2029:
2026:
2020:
2014:
2009:
2005:
2001:
1995:
1989:
1984:
1980:
1976:
1972:
1967:
1964:
1959:
1952:
1948:
1944:
1938:
1932:
1927:
1923:
1915:
1898:
1892:
1887:
1883:
1879:
1873:
1867:
1862:
1858:
1854:
1850:
1845:
1842:
1837:
1830:
1826:
1818:
1815:
1814:
1799:
1796:
1793:
1790:
1787:
1784:
1778:
1772:
1767:
1763:
1759:
1753:
1747:
1742:
1738:
1734:
1728:
1725:
1722:
1716:
1711:
1707:
1703:
1697:
1691:
1686:
1682:
1674:
1656:
1650:
1645:
1641:
1637:
1631:
1625:
1620:
1616:
1612:
1606:
1603:
1597:
1592:
1588:
1579:
1576:
1575:
1571:
1568:
1566:
1565:
1562:
1548:
1543:
1540:
1535:
1531:
1526:
1522:
1519:
1499:
1494:
1491:
1486:
1482:
1477:
1473:
1470:
1442:
1440:
1438:
1434:
1430:
1424:
1416:
1400:
1395:
1392:
1389:
1385:
1381:
1376:
1366:
1362:
1338:
1328:
1324:
1316:
1312:
1305:
1301:
1296:
1288:
1287:
1286:
1276:
1271:
1253:
1228:
1224:
1220:
1215:
1211:
1207:
1205:
1198:
1190:
1187:
1184:
1172:
1169:
1166:
1162:
1158:
1156:
1149:
1139:
1135:
1122:
1118:
1114:
1109:
1105:
1101:
1099:
1092:
1089:
1086:
1082:
1070:
1069:
1068:
1066:
1060:
1052:
1050:
1033:
1030:
1027:
1021:
1018:
998:
995:
992:
989:
986:
983:
980:
977:
971:
968:
962:
959:
950:
936:
933:
930:
910:
907:
904:
901:
881:
858:
855:
852:
849:
846:
839:
838:
837:
823:
815:
799:
796:
793:
790:
787:
782:
778:
774:
771:
768:
763:
759:
749:
747:
743:
738:
736:
732:
728:
724:
718:
710:
708:
691:
688:
685:
676:
673:
670:
664:
659:
655:
651:
646:
642:
619:
615:
611:
608:
605:
602:
599:
594:
590:
586:
581:
573:
570:
567:
556:
540:
537:
531:
528:
522:
519:
499:
496:
493:
490:
487:
477:
469:
464:
462:
460:
456:
444:
428:
425:
422:
419:
414:
410:
406:
403:
400:
395:
391:
368:
364:
360:
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301:
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290: =
289:
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281:
277:
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269:
265:
262:
259:relating one
258:
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250:
230:
227:
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216:
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197:
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164:
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3662:Arto Salomaa
3647:
3638:
3621:
3597:
3590:
3565:
3561:
3551:
3526:
3518:
3507:. Retrieved
3503:
3493:
3482:. Retrieved
3478:
3469:
3458:. Retrieved
3455:www.math.com
3454:
3445:
3434:. Retrieved
3430:
3421:
3410:. Retrieved
3406:
3397:
3385:
3373:
3351:Downing 2003
3347:Shirali 2002
3341:
3297:
3188:
2923:
2917:
2851:of the form
2843:
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2730:
2726:
2721:
2716:
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271:
270:, such that
267:
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252:
246:
135:lies on the
73:, the point
3568:(34): 189.
3533:McGraw-Hill
2816:. In fact,
1275:associative
1252:commutative
742:integration
455:equals sign
449:instead of
249:mathematics
137:unit circle
3790:Categories
3509:2019-12-01
3504:Purplemath
3484:2019-12-01
3460:2019-12-01
3436:2019-12-01
3412:2019-12-01
3328:References
1437:logarithms
816:values of
474:See also:
461:equality.
443:triple bar
3365:Citations
3300:equations
3271:∗
3252:∗
3239:∗
3230:∗
3212:∗
3203:∗
3162:∗
3150:∀
3134:∗
3122:∀
3112:∗
3103:∗
3085:∗
3076:∗
3052:∀
3008:…
2992:∀
2956:…
2875:…
2859:∀
2770:
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2033:−
2015:
2002:−
1990:
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1868:
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1726:⋅
1717:
1692:
1651:
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1492:
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1393:⋅
1221:⋅
1188:⋅
1170:⋅
1115:⋅
1022:
999:θ
996:
990:θ
987:
972:θ
963:
931:θ
902:θ
882:θ
853:θ
850:
824:θ
791:θ
788:
772:θ
769:
689:−
652:−
529:−
423:θ
420:
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401:
304:functions
280:variables
225:θ
222:
206:θ
203:
120:θ
117:
108:θ
105:
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3777:Archived
3664:(eds.).
3306:See also
2543:Typical
1816:quotient
1572:Example
1433:log laws
1266:whereas
735:triangle
257:equality
253:identity
185:. Thus,
3692:Sources
3637:(ed.).
3582:3602492
3546:, p. 21
2849:formula
1577:product
1569:Formula
1512:and/or
1065:integer
555:algebra
445:symbol
3749:
3728:
3709:
3676:
3605:
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3039:monoid
3035:axioms
2915:where
1262:, but
727:angles
453:, the
255:is an
3670:EATCS
3578:JSTOR
3333:Notes
3037:of a
2935:than
2929:terms
2058:power
1268:3 = 9
1264:2 = 8
251:, an
38:, or
3747:ISBN
3726:ISBN
3707:ISBN
3674:ISBN
3603:ISBN
3537:ISBN
2927:are
2921:and
2828:The
2823:even
2430:and
2375:1000
2344:1000
2246:root
1281:and
1258:and
814:real
634:and
512:and
383:and
298:and
274:and
3570:doi
2761:log
2676:log
2652:log
2621:log
2597:log
2569:log
2503:log
2479:log
2451:log
2394:1.5
2363:log
2332:log
2286:log
2257:log
2206:log
2171:log
2146:log
2105:log
2070:log
2006:log
1981:log
1949:log
1924:log
1884:log
1859:log
1827:log
1764:log
1739:log
1708:log
1698:243
1683:log
1642:log
1617:log
1589:log
1532:log
1483:log
1431:or
1019:tan
993:cos
984:sin
960:sin
847:cos
779:cos
760:sin
411:sin
392:cos
247:In
213:sin
194:cos
114:sin
102:cos
3792::
3668:.
3660:;
3656:;
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3576:.
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