39:@0.942772:0.974518:0.963274:0.974518:0.963274:0.956333:0.942772:0.956333:0.000000:0.000000 Pensamiento numérico:@0.721207:0.043660:0.926911:0.043660:0.926911:0.024915:0.721207:0.024915:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 7.1 La logaritmación como operación inversa de la potenciación:@0.073089:0.224215:0.573054:0.224215:0.573054:0.207268:0.073089:0.207268:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ¿Qué son los logaritmos?:@0.073089:0.245863:0.254540:0.245863:0.254540:0.229429:0.073089:0.229429:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Piensa en la siguiente situación: sabemos que 3 elevado a la 4 potencia es 81 y que lo :@0.073089:0.267511:0.710646:0.267511:0.710646:0.251077:0.073089:0.251077:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 podemos escribir en forma exponencial como 3 81. :@0.073089:0.284153:0.470865:0.284147:0.470865:0.267713:0.073089:0.267719:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4:@0.420936:0.278083:0.426173:0.278083:0.426173:0.268502:0.420936:0.268502:0.000000 5:@0.428346:0.283592:0.443679:0.283592:0.443679:0.269876:0.428346:0.269876:0.000000 Vamos a suponer que nos preguntan: ¿a qué potencia debo elevar 3 para obtener 81?, :@0.073089:0.305792:0.710735:0.305792:0.710735:0.289359:0.073089:0.289359:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 lo que podemos escribir de forma exponencial como 3 81 y concluir que el valor de :@0.073089:0.322434:0.710563:0.322434:0.710563:0.306000:0.073089:0.306000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.475961:0.316370:0.480564:0.316370:0.480564:0.306789:0.475961:0.306789:0.000000 5:@0.482791:0.321880:0.498125:0.321880:0.498125:0.308164:0.482791:0.308164:0.000000 x:@0.073089:0.339076:0.080544:0.339076:0.080544:0.322670:0.073089:0.322670:0.000000 es 4. :@0.080470:0.339076:0.120635:0.339076:0.120635:0.322642:0.080470:0.322642:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La expresión 3 81 la notamos o expresamos de la forma logarítmica, así::@0.073089:0.360724:0.610408:0.360721:0.610408:0.344287:0.073089:0.344291:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4:@0.177120:0.354657:0.182357:0.354657:0.182357:0.345076:0.177120:0.345076:0.000000 5:@0.184532:0.360166:0.199865:0.360166:0.199865:0.346450:0.184532:0.346450:0.000000 log 81 4 que se lee “logaritmo en base 3 de 81 es igual a 4”.:@0.073089:0.382366:0.513172:0.382366:0.513172:0.365933:0.073089:0.365933:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.096556:0.384278:0.101792:0.384278:0.101792:0.374697:0.096556:0.374697:0.000000 5:@0.121770:0.381812:0.137103:0.381812:0.137103:0.368096:0.121770:0.368096:0.000000 Podemos afirmar entonces que: log 81 4 porque 3 81:@0.073089:0.404011:0.495382:0.404011:0.495382:0.387577:0.073089:0.387577:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.330854:0.405922:0.336091:0.405922:0.336091:0.396341:0.330854:0.396341:0.000000 5:@0.356069:0.403456:0.371402:0.403456:0.371402:0.389740:0.356069:0.389740:0.000000 4:@0.452612:0.397947:0.457849:0.397947:0.457849:0.388366:0.452612:0.388366:0.000000 5:@0.460022:0.403456:0.475355:0.403456:0.475355:0.389740:0.460022:0.389740:0.000000 Ejemplo 1.:@0.073089:0.533881:0.152608:0.533881:0.152608:0.517212:0.073089:0.517212:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A qué potencia debo elevar 5 para obtener 125.:@0.152498:0.533881:0.513746:0.533881:0.513746:0.517448:0.152498:0.517448:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.271519:0.533881:0.278974:0.533881:0.278974:0.517475:0.271519:0.517475:0.000000 Solución:@0.073089:0.555530:0.137934:0.555530:0.137934:0.538860:0.073089:0.538860:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : lo que tenemos que hacer es hallar = log 25, de donde se concluye que = 3 :@0.137643:0.555530:0.710382:0.555527:0.710382:0.539093:0.137643:0.539096:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.398437:0.555530:0.405892:0.555530:0.405892:0.539124:0.398437:0.539124:0.000000 5:@0.443713:0.558700:0.448950:0.558700:0.448950:0.549118:0.443713:0.549118:0.000000 x:@0.671248:0.555527:0.678703:0.555527:0.678703:0.539121:0.671248:0.539121:0.000000 porque 5 125:@0.073086:0.572169:0.188864:0.572169:0.188864:0.555735:0.073086:0.555735:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.137844:0.566105:0.143081:0.566105:0.143081:0.556524:0.137844:0.556524:0.000000 5:@0.145147:0.571614:0.160480:0.571614:0.160480:0.557899:0.145147:0.557899:0.000000 Ejemplo 2.:@0.073089:0.593813:0.152608:0.593813:0.152608:0.577144:0.073089:0.577144:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Determina los siguiente logaritmos::@0.152498:0.593813:0.414470:0.593813:0.414470:0.577380:0.152498:0.577380:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a. log 32 :@0.073089:0.615459:0.156098:0.615459:0.156098:0.599025:0.073089:0.599025:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.126757:0.618631:0.131994:0.618631:0.131994:0.609051:0.126757:0.609051:0.000000 Solución: :@0.073089:0.637104:0.146788:0.637104:0.146788:0.620434:0.073089:0.620434:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 log 32 5 porque 2 32:@0.146681:0.637104:0.338619:0.637104:0.338619:0.620670:0.146681:0.620670:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.170141:0.640276:0.175378:0.640276:0.175378:0.630695:0.170141:0.630695:0.000000 5:@0.199306:0.636549:0.214639:0.636549:0.214639:0.622833:0.199306:0.622833:0.000000 5:@0.295849:0.631040:0.301086:0.631040:0.301086:0.621459:0.295849:0.621459:0.000000 5:@0.303259:0.636549:0.318592:0.636549:0.318592:0.622833:0.303259:0.622833:0.000000 b. log 256 :@0.073089:0.658749:0.165007:0.658749:0.165007:0.642316:0.073089:0.642316:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4:@0.126757:0.661922:0.131994:0.661922:0.131994:0.652341:0.126757:0.652341:0.000000 Solución::@0.073089:0.680394:0.142959:0.680394:0.142959:0.663724:0.073089:0.663724:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 log 256 4 porque 4 256:@0.142852:0.680394:0.356556:0.680394:0.356556:0.663960:0.142852:0.663960:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4:@0.170266:0.683566:0.175503:0.683566:0.175503:0.673985:0.170266:0.673985:0.000000 5:@0.208334:0.679839:0.223667:0.679839:0.223667:0.666123:0.208334:0.666123:0.000000 4:@0.304877:0.674330:0.310114:0.674330:0.310114:0.664749:0.304877:0.664749:0.000000 5:@0.312287:0.679839:0.327620:0.679839:0.327620:0.666123:0.312287:0.666123:0.000000 7.2 Propiedades de los logaritmos:@0.073089:0.702040:0.339162:0.702040:0.339162:0.685093:0.073089:0.685093:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1. Logaritmo de un producto:@0.073089:0.723684:0.297045:0.723684:0.297045:0.707250:0.073089:0.707250:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 log (x y ) log x log :@0.073084:0.745332:0.277569:0.745330:0.277569:0.728896:0.073084:0.728898:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.096556:0.748502:0.101428:0.748502:0.101428:0.738921:0.096556:0.738921:0.000000 3:@0.120165:0.744775:0.135498:0.744775:0.135498:0.731059:0.120165:0.731059:0.000000 5:@0.156613:0.744775:0.171946:0.744775:0.171946:0.731059:0.156613:0.731059:0.000000 a:@0.197541:0.748502:0.202413:0.748502:0.202413:0.738921:0.197541:0.738921:0.000000 1:@0.216353:0.744775:0.231686:0.744775:0.231686:0.731059:0.216353:0.731059:0.000000 a:@0.257281:0.748502:0.262153:0.748502:0.262153:0.738921:0.257281:0.738921:0.000000 y:@0.266064:0.745330:0.273538:0.745330:0.273538:0.728924:0.266064:0.728924:0.000000 Ejemplo 3.:@0.073101:0.766978:0.152620:0.766978:0.152620:0.750308:0.073101:0.750308:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Prueba que log (9 27) log 9 log 27 :@0.152510:0.766978:0.467798:0.766974:0.467798:0.750540:0.152510:0.750544:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.267266:0.770147:0.272503:0.770147:0.272503:0.760566:0.267266:0.760566:0.000000 3:@0.292325:0.766420:0.307658:0.766420:0.307658:0.752704:0.292325:0.752704:0.000000 5:@0.334610:0.766420:0.349943:0.766420:0.349943:0.752704:0.334610:0.752704:0.000000 3:@0.375537:0.770147:0.380774:0.770147:0.380774:0.760566:0.375537:0.760566:0.000000 1:@0.395800:0.766420:0.411134:0.766420:0.411134:0.752704:0.395800:0.752704:0.000000 3:@0.436728:0.770147:0.441965:0.770147:0.441965:0.760566:0.436728:0.760566:0.000000 log (243 ) 2 3:@0.073091:0.788622:0.203204:0.788620:0.203204:0.772186:0.073091:0.772189:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.096555:0.791792:0.101792:0.791792:0.101792:0.782211:0.096555:0.782211:0.000000 5 1:@0.148168:0.788065:0.192086:0.788065:0.192086:0.774350:0.148168:0.774350:0.000000:0.000000:0.000000 5 5:@0.073089:0.810264:0.110662:0.810264:0.110662:0.793831:0.073089:0.793831:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.084211:0.809710:0.099544:0.809710:0.099544:0.795994:0.084211:0.795994:0.000000 2. Logaritmo de un cociente:@0.073089:0.831910:0.291666:0.831910:0.291666:0.815476:0.073089:0.815476:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 log (x ) log x log :@0.073084:0.853558:0.283878:0.853555:0.283878:0.837121:0.073084:0.837125:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.096555:0.856727:0.101427:0.856727:0.101427:0.847146:0.096555:0.847146:0.000000 4:@0.120165:0.853000:0.135498:0.853000:0.135498:0.839284:0.120165:0.839284:0.000000 y:@0.137627:0.853555:0.145100:0.853555:0.145100:0.837149:0.137627:0.837149:0.000000 5:@0.157728:0.853000:0.173061:0.853000:0.173061:0.839284:0.157728:0.839284:0.000000 a:@0.200391:0.856727:0.205263:0.856727:0.205263:0.847146:0.200391:0.847146:0.000000 2:@0.220935:0.853000:0.236268:0.853000:0.236268:0.839284:0.220935:0.839284:0.000000 a:@0.263591:0.856727:0.268463:0.856727:0.268463:0.847146:0.263591:0.847146:0.000000 y:@0.272373:0.853555:0.279846:0.853555:0.279846:0.837149:0.272373:0.837149:0.000000 Ejemplo 4.:@0.073096:0.875203:0.152615:0.875203:0.152615:0.858533:0.073096:0.858533:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Prueba que log (625 25) log 625 log 25 :@0.152505:0.875203:0.505138:0.875200:0.505138:0.858767:0.152505:0.858769:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.267266:0.878373:0.272503:0.878373:0.272503:0.868792:0.267266:0.868792:0.000000 4:@0.310130:0.874646:0.325463:0.874646:0.325463:0.860930:0.310130:0.860930:0.000000 5:@0.354143:0.874646:0.369476:0.874646:0.369476:0.860930:0.354143:0.860930:0.000000 5:@0.395071:0.878373:0.400308:0.878373:0.400308:0.868792:0.395071:0.868792:0.000000 2:@0.433139:0.874646:0.448472:0.874646:0.448472:0.860930:0.433139:0.860930:0.000000 5:@0.474066:0.878373:0.479303:0.878373:0.479303:0.868792:0.474066:0.868792:0.000000 log 25 4 2:@0.073083:0.896849:0.180755:0.896845:0.180755:0.880411:0.073083:0.880415:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.096556:0.900018:0.101792:0.900018:0.101792:0.890437:0.096556:0.890437:0.000000 5 2:@0.125721:0.896290:0.169637:0.896290:0.169637:0.882575:0.125721:0.882575:0.000000:0.000000:0.000000 2 2:@0.073089:0.918490:0.110662:0.918490:0.110662:0.902056:0.073089:0.902056:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.084211:0.917935:0.099544:0.917935:0.099544:0.904220:0.084211:0.904220:0.000000 Tema 7. La logaritmación:@0.073089:0.098522:0.444979:0.098522:0.444979:0.067281:0.073089:0.067281:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ¿Qué polinomio aritmético representa el área de la figura?:@0.293660:0.172575:0.713152:0.172575:0.713152:0.156141:0.293660:0.156141:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Dados los números , y , diremos que el logaritmo de en base es el número :@0.081796:0.426815:0.689853:0.426815:0.689853:0.410381:0.081796:0.410381:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b x:@0.228013:0.426815:0.278541:0.426815:0.278541:0.410409:0.228013:0.410409:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.500385:0.426815:0.509644:0.426815:0.509644:0.410409:0.500385:0.410409:0.000000 b:@0.575984:0.426815:0.585243:0.426815:0.585243:0.410409:0.575984:0.410409:0.000000 x :@0.690164:0.426815:0.700932:0.426815:0.700932:0.410409:0.690164:0.410409:0.000000:0.000000 siempre que se cumpla que elevado a la sea igual al número .:@0.081786:0.443457:0.559822:0.443457:0.559822:0.427023:0.081786:0.427023:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.287505:0.443457:0.296764:0.443457:0.296764:0.427051:0.287505:0.427051:0.000000 x:@0.390623:0.443457:0.398078:0.443457:0.398078:0.427051:0.390623:0.427051:0.000000 a:@0.547434:0.443457:0.556693:0.443457:0.556693:0.427051:0.547434:0.427051:0.000000 De manera más general, tenemos: log = porque = :@0.081768:0.463677:0.518870:0.463673:0.518870:0.447239:0.081768:0.447243:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.364659:0.465585:0.370582:0.465585:0.370582:0.456004:0.364659:0.456004:0.000000 a x:@0.374493:0.463673:0.409228:0.463673:0.409228:0.447267:0.374493:0.447267:0.000000:0.000000:0.000000:0.000000 b:@0.486225:0.463673:0.495484:0.463673:0.495484:0.447267:0.486225:0.447267:0.000000 x:@0.495427:0.457609:0.500031:0.457609:0.500031:0.448028:0.495427:0.448028:0.000000 a:@0.518777:0.463673:0.528036:0.463673:0.528036:0.447267:0.518777:0.447267:0.000000 Forma:@0.352431:0.495182:0.398786:0.495182:0.398786:0.478748:0.352431:0.478748:0.000000:0.000000:0.000000:0.000000:0.000000 logarítmica:@0.334687:0.511823:0.416591:0.511823:0.416591:0.495390:0.334687:0.495390:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Forma:@0.483951:0.495182:0.530306:0.495182:0.530306:0.478748:0.483951:0.478748:0.000000:0.000000:0.000000:0.000000:0.000000 exponencial:@0.462783:0.511823:0.551432:0.511823:0.551432:0.495390:0.462783:0.495390:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 DBA 9 (Ev. 1) Utiliza las operaciones y sus inversas en problemas de cálculo numérico.:@0.084383:0.971366:0.531658:0.971366:0.531658:0.959414:0.084383:0.959414:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Reto matemático:@0.304294:0.139665:0.426770:0.139665:0.426770:0.124259:0.304294:0.124259:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La logaritmación es una :@0.740294:0.668596:0.901384:0.668596:0.901384:0.653656:0.740294:0.653656:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 herramienta matemática :@0.740294:0.683725:0.907452:0.683725:0.907452:0.668785:0.740294:0.668785:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que nos ayuda a simplificar :@0.740294:0.698854:0.922703:0.698854:0.922703:0.683914:0.740294:0.683914:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 operaciones difíciles. Con :@0.740294:0.713983:0.910966:0.713983:0.910966:0.699043:0.740294:0.699043:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 los logaritmos, podemos :@0.740294:0.729112:0.907150:0.729112:0.907150:0.714172:0.740294:0.714172:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 resolver problemas :@0.740294:0.744240:0.869862:0.744240:0.869862:0.729301:0.740294:0.729301:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 grandes o complicados :@0.740294:0.759369:0.898439:0.759369:0.898439:0.744430:0.740294:0.744430:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de una manera más :@0.740294:0.774498:0.875016:0.774498:0.875016:0.759559:0.740294:0.759559:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 fácil. Son como un truco :@0.740294:0.789627:0.904189:0.789627:0.904189:0.774687:0.740294:0.774687:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemático que convierte :@0.740294:0.804756:0.919520:0.804756:0.919520:0.789816:0.740294:0.789816:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 las multiplicaciones, :@0.740294:0.819885:0.875731:0.819885:0.875731:0.804945:0.740294:0.804945:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 divisiones, potencias y :@0.740294:0.835014:0.890617:0.835014:0.890617:0.820074:0.740294:0.820074:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 raíces en operaciones :@0.740294:0.850143:0.886866:0.850143:0.886866:0.835203:0.740294:0.835203:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 más simples. Es como :@0.740294:0.865272:0.887537:0.865272:0.887537:0.850332:0.740294:0.850332:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 si nos dieran un atajo :@0.740294:0.880401:0.883987:0.880401:0.883987:0.865461:0.740294:0.865461:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 para resolver problemas :@0.740294:0.895530:0.902550:0.895530:0.902550:0.880590:0.740294:0.880590:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 grandes más rápidamente.:@0.740294:0.910659:0.916230:0.910659:0.916230:0.895719:0.740294:0.895719:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Competencia :@0.740294:0.631453:0.838985:0.631453:0.838985:0.616047:0.740294:0.616047:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.740294:0.646582:0.828908:0.646582:0.828908:0.631176:0.740294:0.631176:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Log (8)=3 :@0.808088:0.302093:0.862719:0.302093:0.862719:0.290142:0.808088:0.290142:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.828086:0.304401:0.831895:0.304401:0.831895:0.297433:0.828086:0.297433:0.000000 porque.. 2 =8:@0.808086:0.324918:0.880569:0.324918:0.880569:0.312966:0.808086:0.312966:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3 :@0.860662:0.320508:0.866149:0.320508:0.866149:0.313540:0.860662:0.313540:0.000000:0.000000 4:@0.113912:0.168176:0.121262:0.168176:0.121262:0.154730:0.113912:0.154730:0.000000 2:@0.138551:0.168176:0.145901:0.168176:0.145901:0.154730:0.138551:0.154730:0.000000 1:@0.157528:0.190801:0.164877:0.190801:0.164877:0.177355:0.157528:0.177355:0.000000 2:@0.135690:0.135928:0.143039:0.135928:0.143039:0.122483:0.135690:0.122483:0.000000 3:@0.191444:0.147990:0.198793:0.147990:0.198793:0.134544:0.191444:0.134544:0.000000 MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000