116:@0.054582:0.966558:0.078778:0.966558:0.078778:0.950778:0.054582:0.950778:0.000000:0.000000:0.000000 M.5.1.75. Reconocer la función logarítmica como la función inversa de la función exponencial para calcular el logaritmo de un número, y graficarla analizando esta relación :@0.148948:0.939630:0.894277:0.939630:0.894277:0.929588:0.148948:0.929588:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 para determinar sus características.:@0.148948:0.948432:0.299061:0.948432:0.299061:0.938389:0.148948:0.938389:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Función logarítmica:@0.404235:0.119073:0.650595:0.119073:0.650595:0.074413:0.404235:0.074413:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Sean > 0 con ≠ 1 y la función de en , definida como ( ) = :@0.404236:0.135201:0.895080:0.135201:0.895080:0.118703:0.404236:0.118703:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.441071:0.135316:0.450049:0.135316:0.450049:0.118848:0.441071:0.118848:0.000000 a:@0.510946:0.135316:0.519924:0.135316:0.519924:0.118848:0.510946:0.118848:0.000000 f:@0.562076:0.135316:0.566373:0.135316:0.566373:0.118848:0.562076:0.118848:0.000000 :@0.664351:0.134114:0.677655:0.134114:0.677655:0.121252:0.664351:0.121252:0.000000 :@0.702536:0.134114:0.715841:0.134114:0.715841:0.121252:0.702536:0.121252:0.000000 +:@0.715655:0.127592:0.721956:0.127592:0.721956:0.117974:0.715655:0.117974:0.000000 f x:@0.833946:0.135316:0.851288:0.135316:0.851288:0.118848:0.833946:0.118848:0.000000:0.000000:0.000000 a:@0.875370:0.135316:0.884347:0.135316:0.884347:0.118848:0.875370:0.118848:0.000000 x:@0.884169:0.128916:0.888527:0.128916:0.888527:0.119315:0.884169:0.119315:0.000000 x :@0.404236:0.152667:0.415660:0.152667:0.415660:0.136198:0.404236:0.136198:0.000000:0.000000 :@0.415467:0.152033:0.430119:0.152033:0.430119:0.138660:0.415467:0.138660:0.000000 . Tenemos :@0.429944:0.152551:0.521464:0.152551:0.521464:0.136054:0.429944:0.136054:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000  :@0.433893:0.151465:0.454182:0.151465:0.454182:0.138602:0.433893:0.138602:0.000000:0.000000 Dom f:@0.521272:0.152667:0.565967:0.152667:0.565967:0.136198:0.521272:0.136198:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) = , :@0.555911:0.152551:0.609872:0.152551:0.609872:0.136054:0.555911:0.136054:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.590047:0.151465:0.603352:0.151465:0.603352:0.138602:0.590047:0.138602:0.000000 Rec f:@0.609679:0.152667:0.642989:0.152667:0.642989:0.136198:0.609679:0.136198:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) = . La función es biyectiva.:@0.632932:0.152551:0.852767:0.152551:0.852767:0.136054:0.632932:0.136054:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.667068:0.151465:0.680373:0.151465:0.680373:0.138602:0.667068:0.138602:0.000000 +:@0.680188:0.144942:0.686489:0.144942:0.686489:0.135324:0.680188:0.135324:0.000000 a):@0.404237:0.185966:0.418782:0.185966:0.418782:0.169252:0.404237:0.169252:0.000000:0.000000 Inyectiva: , , ≠ :@0.418782:0.185749:0.604782:0.185744:0.604782:0.169247:0.418782:0.169252:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x x:@0.498367:0.185865:0.525173:0.185860:0.525173:0.169391:0.498367:0.169396:0.000000:0.000000:0.000000 1:@0.505848:0.189022:0.510801:0.189022:0.510801:0.179404:0.505848:0.179404:0.000000 2:@0.525172:0.189022:0.530125:0.189022:0.530125:0.179404:0.525172:0.179404:0.000000 :@0.534267:0.185226:0.548919:0.185226:0.548919:0.171852:0.534267:0.171852:0.000000 :@0.553061:0.184657:0.566366:0.184657:0.566366:0.171795:0.553061:0.171795:0.000000 x:@0.573263:0.185860:0.580738:0.185860:0.580738:0.169391:0.573263:0.169391:0.000000 1:@0.580738:0.189022:0.585691:0.189022:0.585691:0.179404:0.580738:0.179404:0.000000 x:@0.604782:0.185860:0.612257:0.185860:0.612257:0.169391:0.604782:0.169391:0.000000 2 :@0.612256:0.189022:0.619624:0.189022:0.619624:0.179404:0.612256:0.179404:0.000000:0.000000 ⇒:@0.619623:0.184962:0.636985:0.184962:0.636985:0.168790:0.619623:0.168790:0.000000 = ( ) ≠ ( ) = .:@0.636982:0.185744:0.792442:0.185744:0.792442:0.169247:0.636982:0.169247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.641124:0.185860:0.650102:0.185860:0.650102:0.169391:0.641124:0.169391:0.000000 x:@0.650121:0.179460:0.654479:0.179460:0.654479:0.169859:0.650121:0.169859:0.000000 1:@0.654478:0.180088:0.658140:0.180088:0.658140:0.172979:0.654478:0.172979:0.000000 f x:@0.677231:0.185860:0.694955:0.185860:0.694955:0.169391:0.677231:0.169391:0.000000:0.000000:0.000000 1:@0.694954:0.189022:0.699907:0.189022:0.699907:0.179404:0.694954:0.179404:0.000000 f x:@0.724952:0.185860:0.742676:0.185860:0.742676:0.169391:0.724952:0.169391:0.000000:0.000000:0.000000 2:@0.742675:0.189022:0.747628:0.189022:0.747628:0.179404:0.742675:0.179404:0.000000 a:@0.772673:0.185860:0.781651:0.185860:0.781651:0.169391:0.772673:0.169391:0.000000 x:@0.781669:0.179460:0.786027:0.179460:0.786027:0.169859:0.781669:0.169859:0.000000 2:@0.786027:0.180088:0.789688:0.180088:0.789688:0.172979:0.786027:0.172979:0.000000 b):@0.404228:0.203312:0.420237:0.203312:0.420237:0.186597:0.404228:0.186597:0.000000:0.000000 Sobreyectiva: :@0.420237:0.203095:0.525847:0.203095:0.525847:0.186597:0.420237:0.186597:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Rec f:@0.525368:0.203211:0.559448:0.203211:0.559448:0.186742:0.525368:0.186742:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) = , :@0.549199:0.203095:0.696609:0.203095:0.696609:0.186597:0.549199:0.186597:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.583572:0.202008:0.596877:0.202008:0.596877:0.189146:0.583572:0.189146:0.000000 +:@0.596877:0.195486:0.603178:0.195486:0.603178:0.185868:0.596877:0.185868:0.000000 ⇔:@0.606848:0.202312:0.625177:0.202312:0.625177:0.186140:0.606848:0.186140:0.000000 ∀:@0.628847:0.210681:0.640420:0.210681:0.640420:0.191682:0.628847:0.191682:0.000000 y:@0.640427:0.203211:0.648113:0.203211:0.648113:0.186742:0.640427:0.186742:0.000000 :@0.651785:0.202576:0.666437:0.202576:0.666437:0.189203:0.651785:0.189203:0.000000 :@0.670107:0.202008:0.683412:0.202008:0.683412:0.189146:0.670107:0.189146:0.000000 +:@0.683412:0.195486:0.689712:0.195486:0.689712:0.185868:0.683412:0.185868:0.000000 ∃:@0.696137:0.210169:0.706462:0.210169:0.706462:0.192033:0.696137:0.192033:0.000000 x :@0.706461:0.203211:0.718078:0.203211:0.718078:0.186742:0.706461:0.186742:0.000000:0.000000 :@0.717607:0.202576:0.732259:0.202576:0.732259:0.189203:0.717607:0.189203:0.000000 , tal que = ( ) = .:@0.732258:0.203095:0.891770:0.203095:0.891770:0.186597:0.732258:0.186597:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.735929:0.202008:0.749233:0.202008:0.749233:0.189146:0.735929:0.189146:0.000000 y f x:@0.808012:0.203211:0.851570:0.203211:0.851570:0.186742:0.808012:0.186742:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.875671:0.203211:0.884649:0.203211:0.884649:0.186742:0.875671:0.186742:0.000000 x:@0.884658:0.196811:0.889016:0.196811:0.889016:0.187209:0.884658:0.187209:0.000000 La función inversa de la función se define a continuación::@0.404244:0.236293:0.819387:0.236293:0.819387:0.219795:0.404244:0.219795:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.634057:0.236408:0.638353:0.236408:0.638353:0.219940:0.634057:0.219940:0.000000 f:@0.584335:0.271110:0.588631:0.271110:0.588631:0.254641:0.584335:0.254641:0.000000 –1:@0.588626:0.264637:0.599701:0.264637:0.599701:0.255019:0.588626:0.255019:0.000000:0.000000 ::@0.599701:0.270989:0.602456:0.270989:0.602456:0.254492:0.599701:0.254492:0.000000 Esta función está relacionada con la función directa mediante la si-:@0.404236:0.305691:0.891761:0.305691:0.891761:0.289193:0.404236:0.289193:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.778308:0.305806:0.782604:0.305806:0.782604:0.289338:0.778308:0.289338:0.000000 guiente equivalencia::@0.404236:0.323041:0.552056:0.323041:0.552056:0.306544:0.404236:0.306544:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y f x:@0.500774:0.354085:0.545277:0.354085:0.545277:0.337616:0.500774:0.337616:0.000000:0.000000:0.000000:0.000000:0.000000 = ( ) = = ( ), , :@0.508461:0.353969:0.772874:0.353971:0.772874:0.337473:0.508461:0.337472:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.570322:0.354085:0.579300:0.354085:0.579300:0.337616:0.570322:0.337616:0.000000 x:@0.579315:0.347686:0.583673:0.347686:0.583673:0.338084:0.579315:0.338084:0.000000 ⇔:@0.587816:0.353188:0.606144:0.353188:0.606144:0.337016:0.587816:0.337016:0.000000 x f y:@0.610286:0.354087:0.666363:0.354087:0.666363:0.337618:0.610286:0.337618:0.000000:0.000000:0.000000:0.000000:0.000000 –1:@0.641649:0.347618:0.652724:0.347618:0.652724:0.338000:0.641649:0.338000:0.000000:0.000000 x:@0.691639:0.354087:0.699114:0.354087:0.699114:0.337618:0.691639:0.337618:0.000000 :@0.703255:0.353452:0.717907:0.353452:0.717907:0.340079:0.703255:0.340079:0.000000 :@0.722049:0.352884:0.735354:0.352884:0.735354:0.340022:0.722049:0.340022:0.000000 y:@0.742251:0.354087:0.749938:0.354087:0.749938:0.337618:0.742251:0.337618:0.000000 :@0.754078:0.353452:0.768731:0.353452:0.768731:0.340079:0.754078:0.340079:0.000000 :@0.772873:0.352884:0.786177:0.352884:0.786177:0.340022:0.772873:0.340022:0.000000 +:@0.786177:0.346361:0.792478:0.346361:0.792478:0.336743:0.786177:0.336743:0.000000 .:@0.792478:0.353971:0.795233:0.353971:0.795233:0.337473:0.792478:0.337473:0.000000 Al número real ( ) se lo denomina logaritmo de base de y se lo :@0.404245:0.388673:0.895884:0.388673:0.895884:0.372175:0.404245:0.372175:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f y:@0.514615:0.388788:0.544125:0.388788:0.544125:0.372320:0.514615:0.372320:0.000000:0.000000:0.000000 –1:@0.519411:0.382321:0.530485:0.382321:0.530485:0.372703:0.519411:0.372703:0.000000:0.000000 a:@0.800407:0.388788:0.809385:0.388788:0.809385:0.372320:0.800407:0.372320:0.000000 x:@0.835720:0.388788:0.843195:0.388788:0.843195:0.372320:0.835720:0.372320:0.000000 nota :@0.404240:0.406023:0.441750:0.406023:0.441750:0.389526:0.404240:0.389526:0.000000:0.000000:0.000000:0.000000:0.000000 log y:@0.441750:0.406139:0.480672:0.406139:0.480672:0.389670:0.441750:0.389670:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.461799:0.409369:0.467033:0.409369:0.467033:0.399768:0.461799:0.399768:0.000000 ( ). Así,:@0.467033:0.406023:0.518488:0.406023:0.518488:0.389526:0.467033:0.389526:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y a:@0.577538:0.423490:0.613294:0.423490:0.613294:0.407021:0.577538:0.407021:0.000000:0.000000:0.000000 = = :@0.585225:0.423374:0.670842:0.423374:0.670842:0.406876:0.585225:0.406876:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.613304:0.417090:0.617662:0.417090:0.617662:0.407489:0.613304:0.407489:0.000000 ⇔:@0.621805:0.422591:0.640133:0.422591:0.640133:0.406419:0.621805:0.406419:0.000000 x log y:@0.644275:0.423490:0.709770:0.423490:0.709770:0.407021:0.644275:0.407021:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.690895:0.426720:0.696129:0.426720:0.696129:0.417118:0.690895:0.417118:0.000000 ( ).:@0.696130:0.423374:0.718478:0.423374:0.718478:0.406876:0.696130:0.406876:0.000000:0.000000:0.000000:0.000000 Como se ha dicho, la función exponencial dada como :@0.404242:0.455806:0.802513:0.455806:0.802513:0.439308:0.404242:0.439308:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Exp x:@0.803919:0.455921:0.847802:0.455928:0.847802:0.439459:0.803919:0.439452:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.829139:0.459158:0.834373:0.459158:0.834373:0.449557:0.829139:0.449557:0.000000 ( ) = , :@0.834374:0.455812:0.895912:0.455812:0.895912:0.439315:0.834374:0.439315:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.875660:0.455928:0.884637:0.455928:0.884637:0.439459:0.875660:0.439459:0.000000 x:@0.884658:0.449528:0.889016:0.449528:0.889016:0.439927:0.884658:0.439927:0.000000 x:@0.404244:0.473279:0.411719:0.473279:0.411719:0.456810:0.404244:0.456810:0.000000 es biyectiva, por lo tanto, existe la función inversa notada :@0.411719:0.473163:0.895937:0.473163:0.895937:0.456666:0.411719:0.456666:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.418740:0.472645:0.433393:0.472645:0.433393:0.459271:0.418740:0.459271:0.000000 :@0.440421:0.472076:0.453726:0.472076:0.453726:0.459214:0.440421:0.459214:0.000000 como :@0.404234:0.490514:0.450162:0.490514:0.450162:0.474016:0.404234:0.474016:0.000000:0.000000:0.000000:0.000000:0.000000 Log:@0.450162:0.490630:0.473955:0.490630:0.473955:0.474161:0.450162:0.474161:0.000000:0.000000:0.000000 a:@0.474149:0.493859:0.479383:0.493859:0.479383:0.484258:0.474149:0.484258:0.000000 , esto es,:@0.479382:0.490514:0.537504:0.490514:0.537504:0.474016:0.479382:0.474016:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Log:@0.404229:0.523061:0.428021:0.523061:0.428021:0.506592:0.404229:0.506592:0.000000:0.000000:0.000000 a:@0.428221:0.526298:0.433455:0.526298:0.433455:0.516697:0.428221:0.516697:0.000000 ::@0.433454:0.522952:0.436209:0.522952:0.436209:0.506455:0.433454:0.506455:0.000000 :@0.404229:0.540303:0.408371:0.540303:0.408371:0.523805:0.404229:0.523805:0.000000 donde :@0.404229:0.561211:0.455262:0.561211:0.455262:0.544713:0.404229:0.544713:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 log y:@0.455262:0.561326:0.494197:0.561334:0.494197:0.544865:0.455262:0.544858:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.475323:0.564562:0.480557:0.564562:0.480557:0.554961:0.475323:0.554961:0.000000 ( ) es la imagen de la función :@0.480557:0.561218:0.691818:0.561218:0.691818:0.544721:0.480557:0.544721:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Log:@0.691818:0.561334:0.715610:0.561334:0.715610:0.544865:0.691818:0.544865:0.000000:0.000000:0.000000 a:@0.715801:0.564562:0.721035:0.564562:0.721035:0.554961:0.715801:0.554961:0.000000 en .:@0.721034:0.561218:0.757311:0.561218:0.757311:0.544721:0.721034:0.544721:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.746869:0.561334:0.754556:0.561334:0.754556:0.544865:0.746869:0.544865:0.000000 La función :@0.404236:0.594416:0.483376:0.594416:0.483376:0.577918:0.404236:0.577918:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Log:@0.483029:0.594532:0.506820:0.594532:0.506820:0.578063:0.483029:0.578063:0.000000:0.000000:0.000000 a:@0.507027:0.597755:0.512261:0.597755:0.512261:0.588154:0.507027:0.588154:0.000000 se llama función logaritmo de base y el número real :@0.512261:0.594411:0.895851:0.594411:0.895851:0.577913:0.512261:0.577913:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.766001:0.594527:0.774979:0.594527:0.774979:0.578058:0.766001:0.578058:0.000000 log x:@0.404240:0.611877:0.442952:0.611877:0.442952:0.595409:0.404240:0.595409:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.424290:0.615106:0.429524:0.615106:0.429524:0.605505:0.424290:0.605505:0.000000 ( ) con > 0 se llama logaritmo de base de .:@0.429524:0.611762:0.761593:0.611762:0.761593:0.595264:0.429524:0.595264:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.484139:0.611877:0.491614:0.611877:0.491614:0.595409:0.484139:0.595409:0.000000 a:@0.716243:0.611877:0.725220:0.611877:0.725220:0.595409:0.716243:0.595409:0.000000 x:@0.751363:0.611877:0.758838:0.611877:0.758838:0.595409:0.751363:0.595409:0.000000 El :@0.404229:0.644960:0.420450:0.644960:0.420450:0.628462:0.404229:0.628462:0.000000:0.000000:0.000000 Dom Log:@0.420623:0.645075:0.485586:0.645075:0.485586:0.628606:0.420623:0.628606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (:@0.455840:0.644960:0.461793:0.644960:0.461793:0.628462:0.455840:0.628462:0.000000 a:@0.485787:0.648299:0.491021:0.648299:0.491021:0.638697:0.485787:0.638697:0.000000 ) = :@0.491020:0.644955:0.516239:0.644955:0.516239:0.628457:0.491020:0.628457:0.000000:0.000000:0.000000:0.000000 :@0.516415:0.643868:0.529720:0.643868:0.529720:0.631005:0.516415:0.631005:0.000000 +:@0.529720:0.637344:0.536021:0.637344:0.536021:0.627726:0.529720:0.627726:0.000000 , :@0.536021:0.644955:0.542918:0.644955:0.542918:0.628457:0.536021:0.628457:0.000000:0.000000 Rec Log:@0.543091:0.645070:0.596668:0.645070:0.596668:0.628601:0.543091:0.628601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (:@0.566922:0.644955:0.572875:0.644955:0.572875:0.628457:0.566922:0.628457:0.000000 a:@0.596860:0.648299:0.602094:0.648299:0.602094:0.638697:0.596860:0.638697:0.000000 ) = , la función exponencial y la logarít-:@0.602094:0.644955:0.891762:0.644955:0.891762:0.628457:0.602094:0.628457:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.627489:0.643868:0.640793:0.643868:0.640793:0.631005:0.627489:0.631005:0.000000 mica están relacionadas mediante la siguiente equivalencia::@0.404235:0.662305:0.822523:0.662305:0.822523:0.645808:0.404235:0.645808:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y exp x:@0.497653:0.690327:0.567276:0.690333:0.567276:0.673864:0.497653:0.673858:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.505339:0.690211:0.524431:0.690211:0.524431:0.673713:0.505339:0.673713:0.000000:0.000000:0.000000 a:@0.548615:0.693562:0.553849:0.693562:0.553849:0.683960:0.548615:0.683960:0.000000 ( ) = :@0.553848:0.690217:0.626408:0.690217:0.626408:0.673720:0.553848:0.673720:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ⇔:@0.577371:0.689435:0.595699:0.689435:0.595699:0.673263:0.577371:0.673263:0.000000 x log y:@0.599841:0.690333:0.665336:0.690333:0.665336:0.673864:0.599841:0.673864:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.646462:0.693562:0.651696:0.693562:0.651696:0.683960:0.646462:0.683960:0.000000 ( ), , :@0.651696:0.690217:0.775988:0.690217:0.775988:0.673720:0.651696:0.673720:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.694754:0.690333:0.702229:0.690333:0.702229:0.673864:0.694754:0.673864:0.000000 :@0.706369:0.689699:0.721021:0.689699:0.721021:0.676325:0.706369:0.676325:0.000000 :@0.725164:0.689130:0.738468:0.689130:0.738468:0.676268:0.725164:0.676268:0.000000 y:@0.745364:0.690333:0.753050:0.690333:0.753050:0.673864:0.745364:0.673864:0.000000 :@0.757192:0.689699:0.771845:0.689699:0.771845:0.676325:0.757192:0.676325:0.000000 :@0.775987:0.689130:0.789292:0.689130:0.789292:0.676268:0.775987:0.676268:0.000000 +:@0.789292:0.682607:0.795593:0.682607:0.795593:0.672989:0.789292:0.672989:0.000000 .:@0.795592:0.690217:0.798347:0.690217:0.798347:0.673720:0.795592:0.673720:0.000000 Nota que cuando la base = , > 0 se escribe ( ), lo que se denomi-:@0.404238:0.723415:0.891784:0.723415:0.891784:0.706917:0.404238:0.706917:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a e y:@0.580956:0.723531:0.628485:0.723531:0.628485:0.707062:0.580956:0.707062:0.000000:0.000000:0.000000:0.000000:0.000000 ln y:@0.728258:0.723531:0.754673:0.723531:0.754673:0.707062:0.728258:0.707062:0.000000:0.000000:0.000000:0.000000 na logaritmo natural de , o también, logaritmo neperiano. Tenemos::@0.404238:0.740766:0.884191:0.740766:0.884191:0.724268:0.404238:0.724268:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.573235:0.740882:0.580922:0.740882:0.580922:0.724413:0.573235:0.724413:0.000000 y exp x:@0.568570:0.768787:0.632878:0.768787:0.632878:0.752319:0.568570:0.752319:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.576257:0.768672:0.595349:0.768672:0.595349:0.752174:0.576257:0.752174:0.000000:0.000000:0.000000 ( ) = ( ).:@0.619450:0.768672:0.727438:0.768673:0.727438:0.752175:0.619450:0.752174:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ⇔:@0.642971:0.767890:0.661300:0.767890:0.661300:0.751718:0.642971:0.751718:0.000000 x ln y:@0.665442:0.768789:0.718730:0.768789:0.718730:0.752320:0.665442:0.752320:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Cuando = 10, > 0, se escribe :@0.404244:0.801871:0.632302:0.801871:0.632302:0.785373:0.404244:0.785373:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.465411:0.801986:0.474389:0.801986:0.474389:0.785518:0.465411:0.785518:0.000000 y:@0.517369:0.801986:0.525056:0.801986:0.525056:0.785518:0.517369:0.785518:0.000000 log y:@0.632302:0.801986:0.665808:0.801986:0.665808:0.785518:0.632302:0.785518:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) al logaritmo de base 10 de . :@0.652168:0.801871:0.885544:0.801871:0.885544:0.785373:0.652168:0.785373:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.866818:0.801986:0.874505:0.801986:0.874505:0.785518:0.866818:0.785518:0.000000 Ejercicios resueltos:@0.404244:0.836948:0.534899:0.836948:0.534899:0.820161:0.404244:0.820161:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1.:@0.404244:0.854140:0.416516:0.854140:0.416516:0.837425:0.404244:0.837425:0.000000:0.000000 La función :@0.416516:0.853923:0.507467:0.853923:0.507467:0.837425:0.416516:0.837425:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Exp:@0.507467:0.854039:0.532608:0.854039:0.532608:0.837570:0.507467:0.837570:0.000000:0.000000:0.000000 2:@0.532606:0.857195:0.537560:0.857195:0.537560:0.847577:0.532606:0.847577:0.000000 : :@0.537560:0.853918:0.681144:0.853918:0.681144:0.837420:0.537560:0.837420:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.404245:0.888619:0.408387:0.888619:0.408387:0.872122:0.404245:0.872122:0.000000 Su función inversa está definida como :@0.427979:0.888619:0.702812:0.888619:0.702812:0.872122:0.427979:0.872122:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Log:@0.702812:0.888735:0.726604:0.888735:0.726604:0.872266:0.702812:0.872266:0.000000:0.000000:0.000000 2:@0.726606:0.891896:0.731559:0.891896:0.731559:0.882278:0.726606:0.882278:0.000000 : :@0.731558:0.888619:0.829580:0.888619:0.829580:0.872122:0.731558:0.872122:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.404241:0.923321:0.408383:0.923321:0.408383:0.906823:0.404241:0.906823:0.000000 El número real :@0.427976:0.923321:0.534513:0.923321:0.534513:0.906823:0.427976:0.906823:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 log x:@0.534513:0.923437:0.572758:0.923436:0.572758:0.906968:0.534513:0.906968:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.554378:0.926598:0.559331:0.926598:0.559331:0.916980:0.554378:0.916980:0.000000 ( ) se llama logaritmo de base 2 de con > 0. :@0.559330:0.923321:0.895062:0.923321:0.895062:0.906823:0.559330:0.906823:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.810391:0.923436:0.817866:0.923436:0.817866:0.906968:0.810391:0.906968:0.000000 x:@0.853102:0.923436:0.860577:0.923436:0.860577:0.906968:0.853102:0.906968:0.000000 :@0.621073:0.260347:0.634377:0.260347:0.634377:0.247485:0.621073:0.247485:0.000000 +:@0.634377:0.253825:0.640678:0.253825:0.640678:0.244207:0.634377:0.244207:0.000000 :@0.640678:0.261434:0.665106:0.261434:0.665106:0.244936:0.640678:0.244936:0.000000:0.000000:0.000000 →:@0.644820:0.269533:0.660964:0.269533:0.660964:0.249670:0.644820:0.249670:0.000000 :@0.665105:0.260347:0.678409:0.260347:0.678409:0.247485:0.665105:0.247485:0.000000 y:@0.621073:0.282465:0.628759:0.282465:0.628759:0.265996:0.621073:0.265996:0.000000 ( ) = :@0.628759:0.282349:0.707240:0.282349:0.707240:0.265851:0.628759:0.265851:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 →:@0.632901:0.290448:0.649046:0.290448:0.649046:0.270585:0.632901:0.270585:0.000000 f y:@0.653188:0.282465:0.682195:0.282465:0.682195:0.265996:0.653188:0.265996:0.000000:0.000000:0.000000 –1:@0.657480:0.275997:0.668555:0.275997:0.668555:0.266379:0.657480:0.266379:0.000000:0.000000 x.:@0.707240:0.282465:0.717682:0.282465:0.717682:0.265996:0.707240:0.265996:0.000000:0.000000 :@0.459419:0.513610:0.472723:0.513610:0.472723:0.500748:0.459419:0.500748:0.000000 +:@0.472723:0.507087:0.479024:0.507087:0.479024:0.497468:0.472723:0.497468:0.000000 :@0.479024:0.514697:0.503452:0.514697:0.503452:0.498199:0.479024:0.498199:0.000000:0.000000:0.000000 →:@0.483166:0.522796:0.499310:0.522796:0.499310:0.502933:0.483166:0.502933:0.000000 :@0.503451:0.513610:0.516755:0.513610:0.516755:0.500748:0.503451:0.500748:0.000000 y:@0.459419:0.535727:0.467105:0.535727:0.467105:0.519258:0.459419:0.519258:0.000000 :@0.467105:0.535611:0.491534:0.535611:0.491534:0.519113:0.467105:0.519113:0.000000:0.000000:0.000000 →:@0.471247:0.543710:0.487392:0.543710:0.487392:0.523847:0.471247:0.523847:0.000000 log y:@0.491534:0.535727:0.530459:0.535727:0.530459:0.519258:0.491534:0.519258:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.511586:0.538956:0.516819:0.538956:0.516819:0.529355:0.511586:0.529355:0.000000 ( ),:@0.516819:0.535611:0.539167:0.535611:0.539167:0.519113:0.516819:0.519113:0.000000:0.000000:0.000000:0.000000 :@0.558347:0.844825:0.571651:0.844825:0.571651:0.831962:0.558347:0.831962:0.000000 :@0.571650:0.845911:0.596078:0.845911:0.596078:0.829414:0.571650:0.829414:0.000000:0.000000:0.000000 →:@0.575792:0.854010:0.591936:0.854010:0.591936:0.834148:0.575792:0.834148:0.000000 :@0.596076:0.844825:0.609381:0.844825:0.609381:0.831962:0.596076:0.831962:0.000000 +:@0.609381:0.838301:0.615682:0.838301:0.615682:0.828683:0.609381:0.828683:0.000000 x:@0.558347:0.863378:0.565821:0.863378:0.565821:0.846909:0.558347:0.846909:0.000000 :@0.565821:0.863262:0.590250:0.863262:0.590250:0.846765:0.565821:0.846765:0.000000:0.000000:0.000000 →:@0.569963:0.871361:0.586108:0.871361:0.586108:0.851498:0.569963:0.851498:0.000000 exp x:@0.590250:0.863378:0.632728:0.863378:0.632728:0.846909:0.590250:0.846909:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.614346:0.866539:0.619300:0.866539:0.619300:0.856921:0.614346:0.856921:0.000000 ( ) = 2 es:@0.619300:0.863262:0.690074:0.863262:0.690074:0.846765:0.619300:0.846765:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.666267:0.856977:0.670625:0.856977:0.670625:0.847375:0.666267:0.847375:0.000000 :@0.747649:0.880385:0.760953:0.880385:0.760953:0.867523:0.747649:0.867523:0.000000 +:@0.760952:0.873863:0.767253:0.873863:0.767253:0.864245:0.760952:0.864245:0.000000 :@0.767252:0.881472:0.791681:0.881472:0.791681:0.864974:0.767252:0.864974:0.000000:0.000000:0.000000 →:@0.771394:0.889571:0.787539:0.889571:0.787539:0.869708:0.771394:0.869708:0.000000 :@0.791679:0.880385:0.804984:0.880385:0.804984:0.867523:0.791679:0.867523:0.000000 x:@0.747649:0.898938:0.755124:0.898938:0.755124:0.882469:0.747649:0.882469:0.000000 :@0.755124:0.898822:0.779552:0.898822:0.779552:0.882325:0.755124:0.882325:0.000000:0.000000:0.000000 →:@0.759266:0.906921:0.775410:0.906921:0.775410:0.887059:0.759266:0.887059:0.000000 log x:@0.779552:0.898938:0.817792:0.898938:0.817792:0.882469:0.779552:0.882469:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.799410:0.902100:0.804363:0.902100:0.804363:0.892482:0.799410:0.892482:0.000000 ( ):@0.804364:0.898822:0.823745:0.898822:0.823745:0.882325:0.804364:0.882325:0.000000:0.000000:0.000000 Saberes previos:@0.199646:0.115710:0.307805:0.115710:0.307805:0.100238:0.199646:0.100238:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ¿Cómo determinas si :@0.199884:0.144338:0.339476:0.144338:0.339476:0.129274:0.199884:0.129274:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 una función tiene inversa?:@0.153341:0.160180:0.322150:0.160180:0.322150:0.145116:0.153341:0.145116:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Desequilibrio cognitivo:@0.199646:0.207711:0.363493:0.207711:0.363493:0.192238:0.199646:0.192238:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ¿Cómo demuestras que :@0.199884:0.236338:0.357489:0.236338:0.357489:0.221275:0.199884:0.221275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 una función es biyectiva?:@0.153341:0.252180:0.315447:0.252180:0.315447:0.237117:0.153341:0.237117:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Recuerda que…:@0.199644:0.439245:0.309404:0.439245:0.309404:0.423773:0.199644:0.423773:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A B:@0.199880:0.467978:0.224858:0.467978:0.224858:0.452941:0.199880:0.452941:0.000000:0.000000:0.000000 , son dos conjuntos :@0.210171:0.467873:0.352206:0.467873:0.352206:0.452809:0.210171:0.452809:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (no vacíos) cualesquiera y :@0.153337:0.483714:0.331732:0.483714:0.331732:0.468651:0.153337:0.468651:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.324027:0.483820:0.327950:0.483820:0.327950:0.468783:0.324027:0.468783:0.000000 es una función de en que :@0.153337:0.499556:0.347193:0.499556:0.347193:0.484493:0.153337:0.484493:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.272701:0.499662:0.282992:0.499662:0.282992:0.484625:0.272701:0.484625:0.000000 B:@0.306578:0.499662:0.314968:0.499662:0.314968:0.484625:0.306578:0.484625:0.000000 suponemos biyectiva, esto es,:@0.153337:0.515398:0.344518:0.515398:0.344518:0.500335:0.153337:0.500335:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.153337:0.547188:0.157260:0.547188:0.157260:0.532151:0.153337:0.532151:0.000000 : :@0.157260:0.547082:0.163557:0.547082:0.163557:0.532019:0.157260:0.532019:0.000000:0.000000 :@0.153337:0.562924:0.157119:0.562924:0.157119:0.547861:0.153337:0.547861:0.000000 de modo que se verifican las :@0.153337:0.582331:0.341828:0.582331:0.341828:0.567268:0.153337:0.567268:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dos condiciones.:@0.153337:0.598173:0.261600:0.598173:0.261600:0.583110:0.153337:0.583110:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Inyectividad: :@0.153337:0.621342:0.242763:0.621342:0.242763:0.606081:0.153337:0.606081:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x x:@0.153337:0.637091:0.177812:0.637090:0.177812:0.622053:0.153337:0.622055:0.000000:0.000000:0.000000 1:@0.160169:0.639976:0.164691:0.639976:0.164691:0.631195:0.160169:0.631195:0.000000 , , ≠ ( ) ≠ ( ) :@0.164690:0.636985:0.361785:0.636984:0.361785:0.621922:0.164690:0.621921:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.177812:0.639976:0.182335:0.639976:0.182335:0.631195:0.177812:0.631195:0.000000 :@0.186117:0.636485:0.199374:0.636485:0.199374:0.624385:0.186117:0.624385:0.000000 A x:@0.203155:0.637090:0.226567:0.637090:0.226567:0.622053:0.203155:0.622053:0.000000:0.000000:0.000000 1:@0.226567:0.639976:0.231089:0.639976:0.231089:0.631195:0.226567:0.631195:0.000000 x:@0.248522:0.637090:0.255347:0.637090:0.255347:0.622053:0.248522:0.622053:0.000000 2 :@0.255346:0.639976:0.265855:0.639976:0.265855:0.631195:0.255346:0.631195:0.000000:0.000000:0.000000 ⇒:@0.265855:0.636108:0.280736:0.636108:0.280736:0.622246:0.265855:0.622246:0.000000 f x:@0.288296:0.637090:0.304478:0.637090:0.304478:0.622054:0.288296:0.622054:0.000000:0.000000:0.000000 1:@0.304477:0.639976:0.309000:0.639976:0.309000:0.631195:0.304477:0.631195:0.000000 f x:@0.331865:0.637090:0.348047:0.637090:0.348047:0.622054:0.331865:0.622054:0.000000:0.000000:0.000000 2:@0.348047:0.639976:0.352569:0.639976:0.352569:0.631195:0.348047:0.631195:0.000000 Sobreyectividad::@0.153344:0.660152:0.264722:0.660152:0.264722:0.644891:0.153344:0.644891:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Rec f:@0.153344:0.675902:0.184461:0.675902:0.184461:0.660865:0.153344:0.660865:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) = .:@0.175103:0.675796:0.218233:0.675796:0.218233:0.660733:0.175103:0.660733:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 B:@0.207328:0.675902:0.215718:0.675902:0.215718:0.660865:0.207328:0.660865:0.000000 A:@0.176573:0.539627:0.186862:0.539627:0.186862:0.524590:0.176573:0.524590:0.000000 :@0.186862:0.539521:0.209166:0.539521:0.209166:0.524458:0.186862:0.524458:0.000000:0.000000:0.000000 →:@0.190644:0.546915:0.205384:0.546915:0.205384:0.528781:0.190644:0.528781:0.000000 B:@0.209166:0.539627:0.217556:0.539627:0.217556:0.524590:0.209166:0.524590:0.000000 x:@0.176573:0.558715:0.183397:0.558715:0.183397:0.543679:0.176573:0.543679:0.000000 ( ),:@0.183397:0.558610:0.229833:0.558610:0.229833:0.543547:0.183397:0.543547:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 →:@0.187179:0.566004:0.201919:0.566004:0.201919:0.547869:0.187179:0.547869:0.000000 f x:@0.205701:0.558715:0.221882:0.558715:0.221882:0.543679:0.205701:0.543679:0.000000:0.000000:0.000000 Glosario:@0.199644:0.783243:0.267109:0.783243:0.267109:0.765561:0.199644:0.765561:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 biyección: :@0.199880:0.810056:0.270995:0.810056:0.270995:0.794795:0.199880:0.794795:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 aplicación :@0.270995:0.809858:0.340581:0.809858:0.340581:0.794795:0.270995:0.794795:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 biyectiva.:@0.153337:0.825700:0.213882:0.825700:0.213882:0.810637:0.153337:0.810637:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 biyectiva: :@0.153337:0.848869:0.220494:0.848869:0.220494:0.833608:0.153337:0.833608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dicho de una aplica-:@0.220494:0.848671:0.352398:0.848671:0.352398:0.833608:0.220494:0.833608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ción de un conjunto en otro: :@0.153337:0.864513:0.344066:0.864513:0.344066:0.849450:0.153337:0.849450:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que cumple que su correspon-:@0.153337:0.880355:0.352120:0.880355:0.352120:0.865292:0.153337:0.865292:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dencia inversa es también una :@0.153337:0.896197:0.352490:0.896197:0.352490:0.881134:0.153337:0.881134:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 aplicación.:@0.153337:0.912039:0.221516:0.912039:0.221516:0.896976:0.153337:0.896976:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.146908:0.786975:0.158313:0.785466:0.153110:0.763321:0.141706:0.764830:0.000000 c:@0.150290:0.797944:0.160889:0.796542:0.155807:0.774907:0.145207:0.776309:0.000000 b:@0.160665:0.790726:0.173379:0.789044:0.168247:0.767201:0.155534:0.768884:0.000000 biyectiva:@0.690074:0.863479:0.760082:0.863479:0.760082:0.846765:0.690074:0.846765:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . :@0.760084:0.863262:0.766981:0.863262:0.766981:0.846765:0.760084:0.846765:0.000000:0.000000