Sr Examen

Otras calculadoras

Resolver desigualdades/ log3^2(x^2-16)-5log3(x^2-16)>=0

log3^2(x^2-16)-5log3(x^2-16)>=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
                         / 2     \     
   2    / 2     \     log\x  - 16/     
log (3)*\x  - 16/ - 5*------------ >= 0
                         log(3)
(x2−16)log⁡(3)2−5log⁡(x2−16)log⁡(3)≥0\left(x^{2} - 16\right) \log{\left(3 \right)}^{2} - 5 \frac{\log{\left(x^{2} - 16 \right)}}{\log{\left(3 \right)}} \geq 0 
(x^2 - 16)*log(3)^2 - 5*log(x^2 - 16)/log(3) >= 0
Solución detallada
Se da la desigualdad:
(x2−16)log⁡(3)2−5log⁡(x2−16)log⁡(3)≥0\left(x^{2} - 16\right) \log{\left(3 \right)}^{2} - 5 \frac{\log{\left(x^{2} - 16 \right)}}{\log{\left(3 \right)}} \geq 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
(x2−16)log⁡(3)2−5log⁡(x2−16)log⁡(3)=0\left(x^{2} - 16\right) \log{\left(3 \right)}^{2} - 5 \frac{\log{\left(x^{2} - 16 \right)}}{\log{\left(3 \right)}} = 0
Resolvemos:
x1=−−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{1} = - \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x2=−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{2} = \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x3=−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{3} = - \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x4=−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{4} = \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x1=−−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{1} = - \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x2=−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{2} = \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x3=−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{3} = - \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x4=−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{4} = \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
Las raíces dadas
x3=−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{3} = - \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x1=−−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{1} = - \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x2=−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{2} = \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x4=−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x_{4} = \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x3x_{0} \leq x_{3}
Consideremos, por ejemplo, el punto
x0=x3−110x_{0} = x_{3} - \frac{1}{10}
=
−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32+−110- \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}} + - \frac{1}{10}
=
−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32−110- \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}} - \frac{1}{10}
lo sustituimos en la expresión
(x2−16)log⁡(3)2−5log⁡(x2−16)log⁡(3)≥0\left(x^{2} - 16\right) \log{\left(3 \right)}^{2} - 5 \frac{\log{\left(x^{2} - 16 \right)}}{\log{\left(3 \right)}} \geq 0
                                                                         /                                                 2     \     
                                                                         |/            ___________________________________\      |     
                                                                         ||           /      /    3        \              |      |     
        /                                                 2     \        ||          /       |-log (3)     |         3    |      |     
        |/            ___________________________________\      |        ||         /   - 5*W|---------, -1| + 16*log (3) |      |     
        ||           /      /    3        \              |      |        ||  1    \/         \    5        /              |      |     
        ||          /       |-log (3)     |         3    |      |     log||- -- - ----------------------------------------|  - 16|     
        ||         /   - 5*W|---------, -1| + 16*log (3) |      |        ||  10                     3/2                   |      |     
   2    ||  1    \/         \    5        /              |      |        \\                      log   (3)                /      /     
log (3)*||- -- - ----------------------------------------|  - 16| - 5*------------------------------------------------------------ >= 0
        ||  10                     3/2                   |      |                                  1                                   
        \\                      log   (3)                /      /                               log (3)                                

                                                                          /                                                       2\     
                                                                          |      /            ___________________________________\ |     
                                                                          |      |           /      /    3        \              | |     
        /                                                       2\        |      |          /       |-log (3)     |         3    | |     
        |      /            ___________________________________\ |        |      |         /   - 5*W|---------, -1| + 16*log (3) | |     
        |      |           /      /    3        \              | |        |      |  1    \/         \    5        /              | |     
        |      |          /       |-log (3)     |         3    | |   5*log|-16 + |- -- - ----------------------------------------| | >= 0
        |      |         /   - 5*W|---------, -1| + 16*log (3) | |        |      |  10                     3/2                   | |     
   2    |      |  1    \/         \    5        /              | |        \      \                      log   (3)                / /     
log (3)*|-16 + |- -- - ----------------------------------------| | - ---------------------------------------------------------------     
        |      |  10                     3/2                   | |                                log(3)                                 
        \      \                      log   (3)                / /

significa que una de las soluciones de nuestra ecuación será con:
x≤−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x \leq - \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
 _____           _____           _____          
      \         /     \         /
-------•-------•-------•-------•-------
       x3      x1      x2      x4

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≤−−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x \leq - \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x≥−−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32∧x≤−5W(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x \geq - \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}} \wedge x \leq \frac{\sqrt{- 5 W\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
x≥−5W−1(−log⁡(3)35)+16log⁡(3)3log⁡(3)32x \geq \frac{\sqrt{- 5 W_{-1}\left(- \frac{\log{\left(3 \right)}^{3}}{5}\right) + 16 \log{\left(3 \right)}^{3}}}{\log{\left(3 \right)}^{\frac{3}{2}}}
Solución de la desigualdad en el gráfico
0123456-6-5-4-3-2-1-2525
    © Sr Examen – Калькулятор