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Resolver desigualdades/ log^x(x-2)*log^x(x+2)>0

log^x(x-2)*log^x(x+2)>0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
   x           x           
log (x - 2)*log (x + 2) > 0
log⁡(x−2)xlog⁡(x+2)x>0\log{\left(x - 2 \right)}^{x} \log{\left(x + 2 \right)}^{x} > 0 
log(x - 2)^x*log(x + 2)^x > 0
Solución detallada
Se da la desigualdad:
log⁡(x−2)xlog⁡(x+2)x>0\log{\left(x - 2 \right)}^{x} \log{\left(x + 2 \right)}^{x} > 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
log⁡(x−2)xlog⁡(x+2)x=0\log{\left(x - 2 \right)}^{x} \log{\left(x + 2 \right)}^{x} = 0
Resolvemos:
x1=−105.118875071184+1.38646561737862ix_{1} = -105.118875071184 + 1.38646561737862 i
x2=−19.5199290889381+2.31572761056288ix_{2} = -19.5199290889381 + 2.31572761056288 i
x3=−87.1644763425275+1.45832315877969ix_{3} = -87.1644763425275 + 1.45832315877969 i
x4=−25.4608396921231+2.1126228622645ix_{4} = -25.4608396921231 + 2.1126228622645 i
x5=−85.1701319850533+1.46759145157227ix_{5} = -85.1701319850533 + 1.46759145157227 i
x6=−73.2071880583622+1.5304532950715ix_{6} = -73.2071880583622 + 1.5304532950715 i
x7=−15.5681609114383+2.51587862858424ix_{7} = -15.5681609114383 + 2.51587862858424 i
x8=−35.3840488544782+1.89849594655961ix_{8} = -35.3840488544782 + 1.89849594655961 i
x9=−75.200603551132+1.51900165132165ix_{9} = -75.200603551132 + 1.51900165132165 i
x10=−95.1431119565465+1.42403878397291ix_{10} = -95.1431119565465 + 1.42403878397291 i
x11=−81.181861381516+1.48708072811972ix_{11} = -81.181861381516 + 1.48708072811972 i
x12=−13.5968524391923+2.65031823891268ix_{12} = -13.5968524391923 + 2.65031823891268 i
x13=−27.4436555305889+2.06053589621617ix_{13} = -27.4436555305889 + 2.06053589621617 i
x14=−33.3977638274348+1.9335114538576ix_{14} = -33.3977638274348 + 1.9335114538576 i
x15=−61.2509140767952+1.60989349398239ix_{15} = -61.2509140767952 + 1.60989349398239 i
x16=−55.2760562269411+1.65850218787364ix_{16} = -55.2760562269411 + 1.65850218787364 i
x17=−39.3585716423455+1.83653239140527ix_{17} = -39.3585716423455 + 1.83653239140527 i
x18=−23.4791646618538+2.17135018542683ix_{18} = -23.4791646618538 + 2.17135018542683 i
x19=−59.2590172790959+1.62531056284528ix_{19} = -59.2590172790959 + 1.62531056284528 i
x20=−79.1879483623184+1.49734063345215ix_{20} = -79.1879483623184 + 1.49734063345215 i
x21=−83.1759248677926+1.47717109238214ix_{21} = -83.1759248677926 + 1.47717109238214 i
x22=−41.3466997021974+1.80891292936479ix_{22} = -41.3466997021974 + 1.80891292936479 i
x23=−65.2354598407137+1.58111426625679ix_{23} = -65.2354598407137 + 1.58111426625679 i
x24=−101.128270622784+1.40087039633431ix_{24} = -101.128270622784 + 1.40087039633431 i
x25=−77.1941931332931+1.5079728996923ix_{25} = -77.1941931332931 + 1.5079728996923 i
x26=−53.2850287807189+1.67642148574775ix_{26} = -53.2850287807189 + 1.67642148574775 i
x27=−37.3710042670513+1.86629228341476ix_{27} = -37.3710042670513 + 1.86629228341476 i
x28=−89.1589519574163+1.44934902951227ix_{28} = -89.1589519574163 + 1.44934902951227 i
x29=−93.1482748701186+1.43222098184407ix_{29} = -93.1482748701186 + 1.43222098184407 i
x30=−43.3353431915767+1.783181288334ix_{30} = -43.3353431915767 + 1.783181288334 i
x31=−69.2209163999132+1.55474406854811ix_{31} = -69.2209163999132 + 1.55474406854811 i
x32=−49.3039862108778+1.71534551030227ix_{32} = -49.3039862108778 + 1.71534551030227 i
x33=−63.2430664961919+1.59517931382377ix_{33} = -63.2430664961919 + 1.59517931382377 i
x34=−67.2280806150839+1.56765024933939ix_{34} = -67.2280806150839 + 1.56765024933939 i
x35=−45.3244619763413+1.75912513100995ix_{35} = -45.3244619763413 + 1.75912513100995 i
x36=−97.1380598257392+1.41609397763306ix_{36} = -97.1380598257392 + 1.41609397763306 i
x37=−29.4274833006894+2.0138956056109ix_{37} = -29.4274833006894 + 2.0138956056109 i
x38=−21.4987908971154+2.23831762867003ix_{38} = -21.4987908971154 + 2.23831762867003 i
x39=−17.5428825380106+2.40670977713814ix_{39} = -17.5428825380106 + 2.40670977713814 i
x40=−99.1331140928403+1.40837482496147ix_{40} = -99.1331140928403 + 1.40837482496147 i
x41=−9.69133984413244+3.0441032861565ix_{41} = -9.69133984413244 + 3.0441032861565 i
x42=−31.4122161745812+1.97179065963406ix_{42} = -31.4122161745812 + 1.97179065963406 i
x43=−51.2943309868906+1.69533602153123ix_{43} = -51.2943309868906 + 1.69533602153123 i
x44=−91.1535532213376+1.44065319650701ix_{44} = -91.1535532213376 + 1.44065319650701 i
x45=−11.6322287442974+2.82121623594766ix_{45} = -11.6322287442974 + 2.82121623594766 i
x46=−71.2139557420454+1.54235680753043ix_{46} = -71.2139557420454 + 1.54235680753043 i
x47=−2x_{47} = -2
x48=−103.123525511902+1.393570499705ix_{48} = -103.123525511902 + 1.393570499705 i
x49=−47.3140203069425+1.73656445821022ix_{49} = -47.3140203069425 + 1.73656445821022 i
x50=−57.2673921448841+1.64149082188258ix_{50} = -57.2673921448841 + 1.64149082188258 i
Descartamos las soluciones complejas:
x1=−2x_{1} = -2
Las raíces dadas
x1=−2x_{1} = -2
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=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−2+−110-2 + - \frac{1}{10}
=
−2.1-2.1
lo sustituimos en la expresión
log⁡(x−2)xlog⁡(x+2)x>0\log{\left(x - 2 \right)}^{x} \log{\left(x + 2 \right)}^{x} > 0
1log⁡(−2.1−2)2.1log⁡(−2.1+2)2.1>0\frac{1}{\log{\left(-2.1 - 2 \right)}^{2.1} \log{\left(-2.1 + 2 \right)}^{2.1}} > 0
                                               -2.1                              -2.1    
0.0842071065173066*(1 + 0.708723764735013*pi*I)    *(-1 + 0.434294481903252*pi*I)     > 0

Entonces
x<−2x < -2
no se cumple
significa que la solución de la desigualdad será con:
x>−2x > -2
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Solución de la desigualdad en el gráfico
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