150:@0.032380:0.974518:0.061577:0.974518:0.061577:0.956590:0.032380:0.956590:0.000000:0.000000:0.000000 Funciones:@0.073089:0.043661:0.163813:0.043661:0.163813:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5. Considera la ecuación: solución de:@0.293660:0.140040:0.661558:0.140040:0.661558:0.123370:0.293660:0.123370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.496941:0.140040:0.504396:0.140040:0.504396:0.123634:0.496941:0.123634:0.000000 ∈ ⊂ ℝ:@0.508151:0.140622:0.570294:0.140622:0.570294:0.122968:0.508151:0.122968:0.000000:0.000000:0.000000:0.000000:0.000000 I:@0.527221:0.141565:0.534179:0.141565:0.534179:0.122372:0.527221:0.122372:0.000000 ln x:@0.323784:0.168899:0.349205:0.168899:0.349205:0.152493:0.323784:0.152493:0.000000:0.000000:0.000000:0.000000 ( + 10) – :@0.336946:0.168899:0.407685:0.168899:0.407685:0.152466:0.336946:0.152466:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.414619:0.161435:0.423602:0.161435:0.423602:0.145001:0.414619:0.145001:0.000000 10:@0.410128:0.177827:0.428093:0.177827:0.428093:0.161393:0.410128:0.161393:0.000000:0.000000 (2 – 5) = – (α), :@0.430597:0.168910:0.571099:0.168910:0.571099:0.152476:0.430597:0.152476:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ln x:@0.438493:0.168910:0.472805:0.168910:0.472805:0.152503:0.438493:0.152503:0.000000:0.000000:0.000000:0.000000 ln:@0.531376:0.168910:0.544593:0.168910:0.544593:0.152503:0.531376:0.152503:0.000000:0.000000 con α ≥ . Estudia la existencia de soluciones de esta ecuación.:@0.323779:0.207422:0.793667:0.207422:0.793667:0.190988:0.323779:0.190988:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.391844:0.199947:0.400827:0.199947:0.400827:0.183513:0.391844:0.183513:0.000000 2:@0.391844:0.216339:0.400827:0.216339:0.400827:0.199905:0.391844:0.199905:0.000000 Primeramente, determinamos el conjunto en el que esta ecuación tiene sentido. :@0.323793:0.241926:0.930759:0.241926:0.930759:0.225492:0.323793:0.225492:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 I:@0.637968:0.241926:0.641797:0.241926:0.641797:0.225520:0.637968:0.225520:0.000000 Como el dominio de la función logaritmo natural es ]0, ∞[, se sigue que ( + 10) :@0.323793:0.258567:0.930993:0.258567:0.930993:0.242134:0.323793:0.242134:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ln:@0.681133:0.258567:0.698178:0.258567:0.698178:0.242161:0.681133:0.242161:0.000000:0.000000 ln x:@0.860604:0.258567:0.886024:0.258567:0.886024:0.242161:0.860604:0.242161:0.000000:0.000000:0.000000:0.000000 y (2 – 5) están bien definidos, si y solo si se verifica que sus argumentos son :@0.323811:0.275209:0.930663:0.275209:0.930663:0.258775:0.323811:0.258775:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ln x:@0.337525:0.275209:0.372738:0.275209:0.372738:0.258803:0.337525:0.258803:0.000000:0.000000:0.000000:0.000000 positivos, esto es,:@0.323793:0.291851:0.448500:0.291851:0.448500:0.275417:0.323793:0.275417:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.323781:0.315643:0.327812:0.315643:0.327812:0.299209:0.323781:0.299209:0.000000 x:@0.353902:0.315643:0.361357:0.315643:0.361357:0.299237:0.353902:0.299237:0.000000 + 10 > 0, y:@0.361283:0.315643:0.444779:0.315643:0.444779:0.299209:0.361283:0.299209:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2 – 5 > 0:@0.353895:0.335857:0.423935:0.335857:0.423935:0.319424:0.353895:0.319424:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.362804:0.335857:0.370259:0.335857:0.370259:0.319451:0.362804:0.319451:0.000000 Por lo tanto, ( + 10) – (2 – 5) está bien definido, si y solo:@0.323781:0.368115:0.786057:0.368116:0.786057:0.351682:0.323781:0.351681:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ln x:@0.414584:0.368115:0.440004:0.368115:0.440004:0.351709:0.414584:0.351709:0.000000:0.000000:0.000000:0.000000 1:@0.502370:0.359754:0.511353:0.359754:0.511353:0.343320:0.502370:0.343320:0.000000 2:@0.502370:0.376146:0.511353:0.376146:0.511353:0.359712:0.502370:0.359712:0.000000 ln x:@0.523207:0.368116:0.557518:0.368116:0.557518:0.351710:0.523207:0.351710:0.000000:0.000000:0.000000:0.000000 si, > . Así, = , ∞ . Para :@0.323764:0.403411:0.602937:0.403404:0.602937:0.386971:0.323764:0.386977:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.341380:0.403411:0.348835:0.403411:0.348835:0.387005:0.341380:0.387005:0.000000 5:@0.371618:0.395936:0.380601:0.395936:0.380601:0.379502:0.371618:0.379502:0.000000 2:@0.371618:0.412328:0.380601:0.412328:0.380601:0.395894:0.371618:0.395894:0.000000 I :@0.423287:0.404936:0.434166:0.404936:0.434166:0.385743:0.423287:0.385743:0.000000:0.000000 5:@0.465145:0.395936:0.474128:0.395936:0.474128:0.379502:0.465145:0.379502:0.000000 2:@0.465145:0.412328:0.474128:0.412328:0.474128:0.395894:0.465145:0.395894:0.000000 x:@0.568368:0.403404:0.575823:0.403404:0.575823:0.386998:0.568368:0.386998:0.000000 ∈:@0.579707:0.403987:0.595040:0.403987:0.595040:0.386333:0.579707:0.386333:0.000000 5:@0.616856:0.396276:0.625839:0.396276:0.625839:0.379842:0.616856:0.379842:0.000000 2:@0.616856:0.412668:0.625839:0.412668:0.625839:0.396235:0.616856:0.396235:0.000000 , ∞ , tenemos: :@0.629793:0.403404:0.756461:0.403404:0.756461:0.386971:0.629793:0.386971:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ln x:@0.323781:0.439259:0.349202:0.439259:0.349202:0.422853:0.323781:0.422853:0.000000:0.000000:0.000000:0.000000 ( + 10) – (2 – 5) = :@0.336942:0.439259:0.520243:0.439264:0.520243:0.422830:0.336942:0.422825:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.415525:0.431789:0.424507:0.431789:0.424507:0.415355:0.415525:0.415355:0.000000 2:@0.415525:0.448181:0.424507:0.448181:0.424507:0.431747:0.415525:0.431747:0.000000 ln x:@0.436362:0.439264:0.470673:0.439264:0.470673:0.422858:0.436362:0.422858:0.000000:0.000000:0.000000:0.000000 ln :@0.520115:0.439264:0.539810:0.439264:0.539810:0.422858:0.520115:0.422858:0.000000:0.000000:0.000000:0.000000 x:@0.560249:0.430875:0.567704:0.430875:0.567704:0.414469:0.560249:0.414469:0.000000 + 10:@0.567704:0.430875:0.604703:0.430875:0.604703:0.414441:0.567704:0.414441:0.000000:0.000000:0.000000:0.000000:0.000000 2 – 5:@0.559108:0.448529:0.605825:0.448529:0.605825:0.432095:0.559108:0.432095:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.572122:0.448529:0.579577:0.448529:0.579577:0.432123:0.572122:0.432123:0.000000 .:@0.624944:0.439259:0.641234:0.439259:0.641234:0.422853:0.624944:0.422853:0.000000:0.000000:0.000000:0.000000:0.000000 La ecuación propuesta se escribe en forma equivalente, como: :@0.323781:0.470195:0.780126:0.470195:0.780126:0.453761:0.323781:0.453761:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ln :@0.498263:0.502480:0.521199:0.502480:0.521199:0.486074:0.498263:0.486074:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.541636:0.494091:0.549091:0.494091:0.549091:0.477685:0.541636:0.477685:0.000000 + 10:@0.549091:0.494091:0.586089:0.494091:0.586089:0.477657:0.549091:0.477657:0.000000:0.000000:0.000000:0.000000:0.000000 2 – 5:@0.540495:0.511745:0.587212:0.511745:0.587212:0.495311:0.540495:0.495311:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.553509:0.511745:0.560964:0.511745:0.560964:0.495339:0.553509:0.495339:0.000000 = – ln α:@0.606331:0.502475:0.670122:0.502475:0.670122:0.486069:0.606331:0.486069:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ( ) = :@0.655764:0.502475:0.693738:0.502475:0.693738:0.486041:0.655764:0.486041:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ln :@0.693628:0.502475:0.713324:0.502475:0.713324:0.486069:0.693628:0.486069:0.000000:0.000000:0.000000:0.000000 1:@0.723490:0.495719:0.732472:0.495719:0.732472:0.479285:0.723490:0.479285:0.000000 α :@0.723085:0.512360:0.736908:0.512360:0.736908:0.495927:0.723085:0.495927:0.000000:0.000000 ,:@0.742606:0.502475:0.752417:0.502475:0.752417:0.486069:0.742606:0.486069:0.000000:0.000000:0.000000 y por la inyectividad de la función , se obtiene::@0.323781:0.536985:0.674715:0.536985:0.674715:0.520552:0.323781:0.520552:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ln:@0.571138:0.536985:0.588183:0.536985:0.588183:0.520579:0.571138:0.520579:0.000000:0.000000 x:@0.489682:0.560882:0.497137:0.560882:0.497137:0.544475:0.489682:0.544475:0.000000 + 10:@0.497137:0.560882:0.534136:0.560882:0.534136:0.544448:0.497137:0.544448:0.000000:0.000000:0.000000:0.000000:0.000000 2 – 5:@0.488559:0.578536:0.535277:0.578536:0.535277:0.562102:0.488559:0.562102:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.501573:0.578536:0.509028:0.578536:0.509028:0.562130:0.501573:0.562130:0.000000 = :@0.554362:0.569265:0.607340:0.569262:0.607340:0.552828:0.554362:0.552832:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.585748:0.561440:0.594731:0.561440:0.594731:0.545006:0.585748:0.545006:0.000000 α :@0.585343:0.578082:0.599167:0.578082:0.599167:0.561648:0.585343:0.561648:0.000000:0.000000 ⇔:@0.607248:0.568846:0.626428:0.568846:0.626428:0.551857:0.607248:0.551857:0.000000 α( + 10) = 2 – 5 .:@0.626336:0.569262:0.781509:0.569262:0.781509:0.552828:0.626336:0.552828:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.644799:0.569262:0.652254:0.569262:0.652254:0.552856:0.644799:0.552856:0.000000 x:@0.741013:0.569262:0.748468:0.569262:0.748468:0.552856:0.741013:0.552856:0.000000 Elevando al cuadrado ambos miembros en la última igualdad, tenemos::@0.323776:0.607344:0.843580:0.607344:0.843580:0.590910:0.323776:0.590910:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 α ( + 10) = 2 – 5, + 20 + 100 = :@0.441656:0.638533:0.720191:0.638535:0.720191:0.622101:0.441656:0.622099:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.451365:0.632471:0.456602:0.632471:0.456602:0.622890:0.451365:0.622890:0.000000 x:@0.461360:0.638535:0.468815:0.638535:0.468815:0.622128:0.461360:0.622128:0.000000 2:@0.510133:0.632471:0.515370:0.632471:0.515370:0.622890:0.510133:0.622890:0.000000 x:@0.543008:0.638535:0.550463:0.638535:0.550463:0.622128:0.543008:0.622128:0.000000 x:@0.595248:0.638535:0.602703:0.638535:0.602703:0.622128:0.595248:0.622128:0.000000 2:@0.602643:0.632471:0.607879:0.632471:0.607879:0.622890:0.602643:0.622890:0.000000 x:@0.644427:0.638535:0.651882:0.638535:0.651882:0.622128:0.644427:0.622128:0.000000 1:@0.728639:0.631246:0.737621:0.631246:0.737621:0.614812:0.728639:0.614812:0.000000 α:@0.725620:0.647888:0.735412:0.647888:0.735412:0.631454:0.725620:0.631454:0.000000 2:@0.735407:0.641824:0.740644:0.641824:0.740644:0.632243:0.735407:0.632243:0.000000 (2 – 5),:@0.746124:0.638535:0.809058:0.638535:0.809058:0.622101:0.746124:0.622101:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.767716:0.638535:0.775170:0.638535:0.775170:0.622128:0.767716:0.622128:0.000000 x:@0.503074:0.671860:0.510529:0.671860:0.510529:0.655454:0.503074:0.655454:0.000000 2:@0.510477:0.665800:0.515714:0.665800:0.515714:0.656219:0.510477:0.656219:0.000000 + 20 – :@0.515668:0.671864:0.580720:0.671864:0.580720:0.655430:0.515668:0.655430:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.552262:0.671864:0.559717:0.671864:0.559717:0.655458:0.552262:0.655458:0.000000 2:@0.587985:0.664039:0.596968:0.664039:0.596968:0.647606:0.587985:0.647606:0.000000 α:@0.584967:0.680681:0.594759:0.680681:0.594759:0.664247:0.584967:0.664247:0.000000 2:@0.594756:0.674617:0.599993:0.674617:0.599993:0.665036:0.594756:0.665036:0.000000 :@0.599992:0.680524:0.603840:0.680524:0.603840:0.664837:0.599992:0.664837:0.000000 + 100 + :@0.604247:0.671864:0.687908:0.671864:0.687908:0.655430:0.604247:0.655430:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.612144:0.671864:0.619599:0.671864:0.619599:0.655458:0.612144:0.655458:0.000000 5:@0.696275:0.662431:0.705258:0.662431:0.705258:0.645997:0.696275:0.645997:0.000000 α:@0.693257:0.679073:0.703049:0.679073:0.703049:0.662639:0.693257:0.662639:0.000000 2:@0.703044:0.673009:0.708281:0.673009:0.708281:0.663428:0.703044:0.663428:0.000000 :@0.708282:0.678915:0.712130:0.678915:0.712130:0.663228:0.708282:0.663228:0.000000 = 0.:@0.713677:0.671864:0.747631:0.671864:0.747631:0.655430:0.713677:0.655430:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Determinemos las raíces de esta ecuación en términos de α. Para el efecto, aplicamos :@0.323774:0.702804:0.931197:0.702804:0.931197:0.686370:0.323774:0.686370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la conocida fórmula de obtención de las raíces de una ecuación de segundo grado. :@0.323774:0.719445:0.931308:0.719445:0.931308:0.703012:0.323774:0.703012:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Tenemos::@0.323774:0.736087:0.393254:0.736087:0.393254:0.719653:0.323774:0.719653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.323781:0.786688:0.334476:0.786688:0.334476:0.770282:0.323781:0.770282:0.000000:0.000000 = :@0.353902:0.786688:0.372788:0.786688:0.372788:0.770255:0.353902:0.770255:0.000000:0.000000:0.000000 2:@0.539733:0.797395:0.552747:0.797395:0.552747:0.780961:0.539733:0.780961:0.000000:0.000000 2:@0.444727:0.762438:0.453710:0.762438:0.453710:0.746004:0.444727:0.746004:0.000000 α:@0.441708:0.779080:0.451501:0.779080:0.451501:0.762646:0.441708:0.762646:0.000000 2:@0.451496:0.773016:0.456732:0.773016:0.456732:0.763435:0.451496:0.763435:0.000000 :@0.456732:0.778922:0.462319:0.778922:0.462319:0.763235:0.456732:0.763235:0.000000:0.000000 2:@0.560472:0.763331:0.569454:0.763331:0.569454:0.746897:0.560472:0.746897:0.000000 α:@0.557453:0.779973:0.567245:0.779973:0.567245:0.763539:0.557453:0.763539:0.000000 2:@0.567242:0.773909:0.572479:0.773909:0.572479:0.764328:0.567242:0.764328:0.000000 :@0.572478:0.779815:0.576326:0.779815:0.576326:0.764128:0.572478:0.764128:0.000000 5:@0.692798:0.763331:0.701781:0.763331:0.701781:0.746897:0.692798:0.746897:0.000000 α:@0.689779:0.779973:0.699572:0.779973:0.699572:0.763539:0.689779:0.763539:0.000000 2:@0.699569:0.773909:0.704805:0.773909:0.704805:0.764328:0.699569:0.764328:0.000000 :@0.704805:0.779815:0.708652:0.779815:0.708652:0.764128:0.704805:0.764128:0.000000 – 20 – ± 20 – – 4 100 +:@0.376540:0.770835:0.676467:0.770835:0.676467:0.754401:0.376540:0.754401:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.583692:0.757124:0.588928:0.757124:0.588928:0.747543:0.583692:0.747543:0.000000 =:@0.353907:0.839079:0.364804:0.839079:0.364804:0.822645:0.353907:0.822645:0.000000 2:@0.551007:0.848892:0.564021:0.848892:0.564021:0.832459:0.551007:0.832459:0.000000:0.000000 2:@0.432538:0.813934:0.441521:0.813934:0.441521:0.797501:0.432538:0.797501:0.000000 α:@0.429519:0.830576:0.439312:0.830576:0.439312:0.814142:0.429519:0.814142:0.000000 2:@0.439308:0.824512:0.444545:0.824512:0.444545:0.814931:0.439308:0.814931:0.000000 :@0.444544:0.830419:0.448392:0.830419:0.448392:0.814732:0.444544:0.814732:0.000000 – 20 + ± 400 – + – 400 –:@0.369993:0.821437:0.684923:0.821437:0.684923:0.805003:0.369993:0.805003:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 80:@0.548564:0.813934:0.566529:0.813934:0.566529:0.797501:0.548564:0.797501:0.000000:0.000000 α:@0.550036:0.830576:0.559829:0.830576:0.559829:0.814142:0.550036:0.814142:0.000000 2:@0.559824:0.824512:0.565061:0.824512:0.565061:0.814931:0.559824:0.814931:0.000000 :@0.565062:0.830419:0.568910:0.830419:0.568910:0.814732:0.565062:0.814732:0.000000 20:@0.700062:0.813934:0.718028:0.813934:0.718028:0.797501:0.700062:0.797501:0.000000:0.000000 α:@0.701535:0.830576:0.711327:0.830576:0.711327:0.814142:0.701535:0.814142:0.000000 2:@0.711324:0.824512:0.716561:0.824512:0.716561:0.814931:0.711324:0.814931:0.000000 :@0.716560:0.830419:0.720408:0.830419:0.720408:0.814732:0.716560:0.814732:0.000000 4:@0.602069:0.813934:0.611051:0.813934:0.611051:0.797501:0.602069:0.797501:0.000000 α:@0.599050:0.830576:0.608843:0.830576:0.608843:0.814142:0.599050:0.814142:0.000000 4:@0.608837:0.824512:0.614074:0.824512:0.614074:0.814931:0.608837:0.814931:0.000000 :@0.293660:0.893804:0.297691:0.893804:0.297691:0.877370:0.293660:0.877370:0.000000 = = .:@0.353907:0.893804:0.869642:0.893804:0.869642:0.877370:0.353907:0.877370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ⇔:@0.488710:0.325488:0.506147:0.325488:0.506147:0.310044:0.488710:0.310044:0.000000 x:@0.541783:0.311553:0.549238:0.311553:0.549238:0.295147:0.541783:0.295147:0.000000 > –10, y:@0.549238:0.311553:0.614805:0.311553:0.614805:0.295119:0.549238:0.295119:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.541783:0.335337:0.549238:0.335337:0.549238:0.318931:0.541783:0.318931:0.000000 > .:@0.549238:0.335337:0.587599:0.335337:0.587599:0.318903:0.549238:0.318903:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.580607:0.328748:0.589590:0.328748:0.589590:0.312314:0.580607:0.312314:0.000000 2:@0.580607:0.345140:0.589590:0.345140:0.589590:0.328706:0.580607:0.328706:0.000000 2:@0.479259:0.904608:0.492273:0.904608:0.492273:0.888174:0.479259:0.888174:0.000000:0.000000 2:@0.737650:0.904608:0.750664:0.904608:0.750664:0.888174:0.737650:0.888174:0.000000:0.000000 2:@0.438806:0.868757:0.447789:0.868757:0.447789:0.852323:0.438806:0.852323:0.000000 α:@0.435788:0.885399:0.445580:0.885399:0.445580:0.868965:0.435788:0.868965:0.000000 2:@0.445575:0.879335:0.450812:0.879335:0.450812:0.869754:0.445575:0.869754:0.000000 :@0.450813:0.885241:0.454661:0.885241:0.454661:0.869554:0.450813:0.869554:0.000000 2:@0.697062:0.869651:0.706045:0.869651:0.706045:0.853217:0.697062:0.853217:0.000000 α:@0.694043:0.886292:0.703836:0.886292:0.703836:0.869859:0.694043:0.869859:0.000000 2:@0.703831:0.880229:0.709067:0.880229:0.709067:0.870648:0.703831:0.870648:0.000000 :@0.709068:0.886135:0.712916:0.886135:0.712916:0.870448:0.709068:0.870448:0.000000 – 20 + ± –:@0.374364:0.877153:0.542897:0.877153:0.542897:0.860719:0.374364:0.860719:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 – 20 + ±:@0.632600:0.878041:0.730084:0.878041:0.730084:0.861607:0.632600:0.861607:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 100:@0.556717:0.870544:0.583665:0.870544:0.583665:0.854111:0.556717:0.854111:0.000000:0.000000:0.000000 α:@0.562680:0.887186:0.572473:0.887186:0.572473:0.870752:0.562680:0.870752:0.000000 2:@0.572468:0.881122:0.577705:0.881122:0.577705:0.871541:0.572468:0.871541:0.000000 :@0.577706:0.887029:0.581554:0.887029:0.581554:0.871342:0.577706:0.871342:0.000000 4:@0.511485:0.870544:0.520468:0.870544:0.520468:0.854111:0.511485:0.854111:0.000000 α:@0.508466:0.887186:0.518259:0.887186:0.518259:0.870752:0.508466:0.870752:0.000000 4:@0.518254:0.881122:0.523491:0.881122:0.523491:0.871541:0.518254:0.871541:0.000000 4 – 100α:@0.772885:0.870544:0.836242:0.870544:0.836242:0.854111:0.772885:0.854111:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.836242:0.864481:0.841479:0.864481:0.841479:0.854900:0.836242:0.854900:0.000000 α:@0.799667:0.887186:0.809460:0.887186:0.809460:0.870752:0.799667:0.870752:0.000000 4:@0.809460:0.881122:0.814697:0.881122:0.814697:0.871541:0.809460:0.871541:0.000000 Competencia :@0.078581:0.179567:0.187141:0.179567:0.187141:0.162620:0.078581:0.162620:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.078581:0.196208:0.176057:0.196208:0.176057:0.179262:0.078581:0.179262:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Las ecuaciones :@0.082576:0.215807:0.185623:0.215807:0.185623:0.200867:0.082576:0.200867:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 logarítmicas son :@0.082576:0.230936:0.194385:0.230936:0.194385:0.215996:0.082576:0.215996:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 aquellas ecuaciones :@0.082576:0.246065:0.218555:0.246065:0.218555:0.231125:0.082576:0.231125:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en las que la incógnita :@0.082576:0.261194:0.234143:0.261194:0.234143:0.246254:0.082576:0.246254:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 aparece afectada por un :@0.082576:0.276323:0.246516:0.276323:0.246516:0.261383:0.082576:0.261383:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 logaritmo.:@0.082576:0.291451:0.150052:0.291451:0.150052:0.276512:0.082576:0.276512:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Para resolver ecuaciones :@0.082576:0.306580:0.247352:0.306580:0.247352:0.291641:0.082576:0.291641:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 logarítmicas, ten en :@0.082576:0.321709:0.215625:0.321709:0.215625:0.306770:0.082576:0.306770:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cuenta lo siguiente::@0.082576:0.336838:0.211688:0.336838:0.211688:0.321898:0.082576:0.321898:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 • Aplica las propiedades :@0.087319:0.351967:0.258298:0.351967:0.258298:0.337027:0.087319:0.337027:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de los logaritmos :@0.106293:0.367096:0.223842:0.367096:0.223842:0.352156:0.106293:0.352156:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 hasta obtener un solo :@0.106293:0.382225:0.253979:0.382225:0.253979:0.367285:0.106293:0.367285:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 logaritmo en ambos :@0.106293:0.397354:0.244391:0.397354:0.244391:0.382414:0.106293:0.382414:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 miembros de la :@0.106293:0.412483:0.212933:0.412483:0.212933:0.397543:0.106293:0.397543:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ecuación.:@0.106293:0.427612:0.169564:0.427612:0.169564:0.412672:0.106293:0.412672:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 • Resuelve la ecuación :@0.087319:0.442741:0.247045:0.442741:0.247045:0.427801:0.087319:0.427801:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que queda y verifica :@0.106293:0.457870:0.242920:0.457870:0.242920:0.442930:0.106293:0.442930:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la validez de las :@0.106293:0.472999:0.211798:0.472999:0.211798:0.458059:0.106293:0.458059:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 soluciones obtenidas.:@0.106293:0.488128:0.248835:0.488128:0.248835:0.473188:0.106293:0.473188:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Dr. Manuel, Wikimedia Commons:@0.223093:0.782123:0.223093:0.682122:0.211196:0.682122:0.211196:0.782123:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Interdisciplinaria :@0.078581:0.561421:0.201990:0.561421:0.201990:0.546015:0.078581:0.546015:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Matemática y :@0.078581:0.576550:0.171491:0.576550:0.171491:0.561610:0.078581:0.561610:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 estudios sociales:@0.078581:0.591679:0.188941:0.591679:0.188941:0.576739:0.078581:0.576739:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Las ecuaciones :@0.080953:0.619292:0.184000:0.619292:0.184000:0.604352:0.080953:0.604352:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 logarítmicas son :@0.080953:0.634421:0.192762:0.634421:0.192762:0.619481:0.080953:0.619481:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ampliamente utilizadas :@0.080953:0.649550:0.237910:0.649550:0.237910:0.634610:0.080953:0.634610:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 para resolver problemas :@0.080953:0.664679:0.243209:0.664679:0.243209:0.649739:0.080953:0.649739:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de crecimiento :@0.080953:0.679808:0.184174:0.679808:0.184174:0.664868:0.080953:0.664868:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 poblacional, que son :@0.080953:0.694937:0.222081:0.694937:0.222081:0.679997:0.080953:0.679997:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 modelados mediante :@0.080953:0.710065:0.226531:0.710065:0.226531:0.695126:0.080953:0.695126:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ecuaciones exponenciales.:@0.080953:0.725194:0.256938:0.725194:0.256938:0.710255:0.080953:0.710255:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Indaga responde:@0.080953:0.858117:0.209039:0.858117:0.209039:0.842963:0.080953:0.842963:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y :@0.128904:0.858117:0.143462:0.858117:0.143462:0.843177:0.128904:0.843177:0.000000:0.000000:0.000000 con :@0.208942:0.858117:0.241406:0.858117:0.241406:0.843177:0.208942:0.843177:0.000000:0.000000:0.000000:0.000000:0.000000 tus palabras: ¿qué es un :@0.080961:0.873246:0.241606:0.873246:0.241606:0.858306:0.080961:0.858306:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 logaritmo? :@0.080961:0.888375:0.155807:0.888375:0.155807:0.873435:0.080961:0.873435:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Escribe:@0.155678:0.888375:0.204988:0.888375:0.204988:0.873221:0.155678:0.873221:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 otras :@0.204892:0.888375:0.244736:0.888375:0.244736:0.873435:0.204892:0.873435:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 aplicaciones.:@0.080961:0.903504:0.165009:0.903504:0.165009:0.888564:0.080961:0.888564:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 EC. Identifica fenómenos en la física, la ingeniería, la economía u otras ciencias que :@0.479433:0.965242:0.916619:0.965242:0.916619:0.953290:0.479433:0.953290:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 pueden modelarse mediante funciones y ecuaciones exponenciales o logarítmicas.:@0.479433:0.977345:0.917844:0.977345:0.917844:0.965393:0.479433:0.965393:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 MUESTRA EDITORIAL:@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