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lnx/ln2+ln3/lnx*ln3/ln2+ln9/ln2<0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
         log(3)                    
         ------*log(3)             
log(x)   log(x)          log(9)    
------ + ------------- + ------ < 0
log(2)       log(2)      log(2)    
(log(3)log(x)log(3)log(2)+log(x)log(2))+log(9)log(2)<0\left(\frac{\frac{\log{\left(3 \right)}}{\log{\left(x \right)}} \log{\left(3 \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}\right) + \frac{\log{\left(9 \right)}}{\log{\left(2 \right)}} < 0
((log(3)/log(x))*log(3))/log(2) + log(x)/log(2) + log(9)/log(2) < 0
Solución detallada
Se da la desigualdad:
(log(3)log(x)log(3)log(2)+log(x)log(2))+log(9)log(2)<0\left(\frac{\frac{\log{\left(3 \right)}}{\log{\left(x \right)}} \log{\left(3 \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}\right) + \frac{\log{\left(9 \right)}}{\log{\left(2 \right)}} < 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
(log(3)log(x)log(3)log(2)+log(x)log(2))+log(9)log(2)=0\left(\frac{\frac{\log{\left(3 \right)}}{\log{\left(x \right)}} \log{\left(3 \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}\right) + \frac{\log{\left(9 \right)}}{\log{\left(2 \right)}} = 0
Resolvemos:
x1=13x_{1} = \frac{1}{3}
x1=13x_{1} = \frac{1}{3}
Las raíces dadas
x1=13x_{1} = \frac{1}{3}
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x1x_{0} < x_{1}
Consideremos, por ejemplo, el punto
x0=x1110x_{0} = x_{1} - \frac{1}{10}
=
110+13- \frac{1}{10} + \frac{1}{3}
=
730\frac{7}{30}
lo sustituimos en la expresión
(log(3)log(x)log(3)log(2)+log(x)log(2))+log(9)log(2)<0\left(\frac{\frac{\log{\left(3 \right)}}{\log{\left(x \right)}} \log{\left(3 \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}\right) + \frac{\log{\left(9 \right)}}{\log{\left(2 \right)}} < 0
(log(730)log(2)+log(3)log(730)log(3)log(2))+log(9)log(2)<0\left(\frac{\log{\left(\frac{7}{30} \right)}}{\log{\left(2 \right)}} + \frac{\frac{\log{\left(3 \right)}}{\log{\left(\frac{7}{30} \right)}} \log{\left(3 \right)}}{\log{\left(2 \right)}}\right) + \frac{\log{\left(9 \right)}}{\log{\left(2 \right)}} < 0
                            2            
log(9)   log(7/30)       log (3)         
------ + --------- + ---------------- < 0
log(2)     log(2)    log(2)*log(7/30)    
    

significa que la solución de la desigualdad será con:
x<13x < \frac{1}{3}
 _____          
      \    
-------ο-------
       x1
Solución de la desigualdad en el gráfico
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-2000020000
Respuesta rápida [src]
Or(And(0 < x, x < 1/3), And(1/3 < x, x < 1))
(0<xx<13)(13<xx<1)\left(0 < x \wedge x < \frac{1}{3}\right) \vee \left(\frac{1}{3} < x \wedge x < 1\right)
((0 < x)∧(x < 1/3))∨((1/3 < x)∧(x < 1))
Respuesta rápida 2 [src]
(0, 1/3) U (1/3, 1)
x in (0,13)(13,1)x\ in\ \left(0, \frac{1}{3}\right) \cup \left(\frac{1}{3}, 1\right)
x in Union(Interval.open(0, 1/3), Interval.open(1/3, 1))