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Resolver desigualdades/ log3(2^x-1)+log(2^x-1)3<=10/3

log3(2^x-1)+log(2^x-1)3<=10/3 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
   / x    \                        
log\2  - 1/      / x    \          
----------- + log\2  - 1/*3 <= 10/3
   log(3)
log⁡(2x−1)log⁡(3)+3log⁡(2x−1)≤103\frac{\log{\left(2^{x} - 1 \right)}}{\log{\left(3 \right)}} + 3 \log{\left(2^{x} - 1 \right)} \leq \frac{10}{3} 
log(2^x - 1)/log(3) + 3*log(2^x - 1) <= 10/3
Solución detallada
Se da la desigualdad:
log⁡(2x−1)log⁡(3)+3log⁡(2x−1)≤103\frac{\log{\left(2^{x} - 1 \right)}}{\log{\left(3 \right)}} + 3 \log{\left(2^{x} - 1 \right)} \leq \frac{10}{3}
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
log⁡(2x−1)log⁡(3)+3log⁡(2x−1)=103\frac{\log{\left(2^{x} - 1 \right)}}{\log{\left(3 \right)}} + 3 \log{\left(2^{x} - 1 \right)} = \frac{10}{3}
Resolvemos:
x1=log⁡(1+3103(1+3log⁡(3)))log⁡(2)x_{1} = \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}
x1=log⁡(1+3103(1+3log⁡(3)))log⁡(2)x_{1} = \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}
Las raíces dadas
x1=log⁡(1+3103(1+3log⁡(3)))log⁡(2)x_{1} = \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x1x_{0} \leq x_{1}
Consideremos, por ejemplo, el punto
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+log⁡(1+3103(1+3log⁡(3)))log⁡(2)- \frac{1}{10} + \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}
=
−110+log⁡(1+3103(1+3log⁡(3)))log⁡(2)- \frac{1}{10} + \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}
lo sustituimos en la expresión
log⁡(2x−1)log⁡(3)+3log⁡(2x−1)≤103\frac{\log{\left(2^{x} - 1 \right)}}{\log{\left(3 \right)}} + 3 \log{\left(2^{x} - 1 \right)} \leq \frac{10}{3}
log⁡(−1+2−110+log⁡(1+3103(1+3log⁡(3)))log⁡(2))log⁡(3)+3log⁡(−1+2−110+log⁡(1+3103(1+3log⁡(3)))log⁡(2))≤103\frac{\log{\left(-1 + 2^{- \frac{1}{10} + \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}} \right)}}{\log{\left(3 \right)}} + 3 \log{\left(-1 + 2^{- \frac{1}{10} + \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}} \right)} \leq \frac{10}{3}
                                                    /                /            10       \\        
     /                /            10       \\      |                |     ----------------||        
     |                |     ----------------||      |                |     3*(1 + 3*log(3))||        
     |                |     3*(1 + 3*log(3))||      |        1    log\1 + 3                /|        
     |        1    log\1 + 3                /|      |      - -- + --------------------------| <= 10/3
     |      - -- + --------------------------|      |        10             log(2)          |        
     |        10             log(2)          |   log\-1 + 2                                 /        
3*log\-1 + 2                                 / + --------------------------------------------        
                                                                    log(3)

significa que la solución de la desigualdad será con:
x≤log⁡(1+3103(1+3log⁡(3)))log⁡(2)x \leq \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}
 _____          
      \    
-------•-------
       x1
Solución de la desigualdad en el gráfico
01234567-5-4-3-2-1-5050
Respuesta rápida 2 [src] 
       /            10       \ 
       |     ----------------| 
       |     3*(1 + 3*log(3))| 
    log\1 + 3                / 
(0, --------------------------]
              log(2)
x in (0,log⁡(1+3103(1+3log⁡(3)))log⁡(2)]x\ in\ \left(0, \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}}\right] 
x in Interval.Lopen(0, log(1 + 3^(10/(3*(1 + 3*log(3)))))/log(2))
Respuesta rápida [src] 
   /        /            10       \       \
   |        |     ----------------|       |
   |        |     3*(1 + 3*log(3))|       |
   |     log\1 + 3                /       |
And|x <= --------------------------, 0 < x|
   \               log(2)                 /
x≤log⁡(1+3103(1+3log⁡(3)))log⁡(2)∧0<xx \leq \frac{\log{\left(1 + 3^{\frac{10}{3 \left(1 + 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(2 \right)}} \wedge 0 < x 
(0 < x)∧(x <= log(1 + 3^(10/(3*(1 + 3*log(3)))))/log(2))
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