Question

Solve the following inequality:
${\log }_{\sqrt{2}}(5^x-1){\log }_{\sqrt{2}}\frac{2\sqrt{2}}{5^x-1}>2$

Answer 1

${\log }_{\sqrt{2}}(5^x-1){\log }_{\sqrt{2}}\left[\frac{2\sqrt{2}}{(5^x-1)}\right]>2$

${\log }_{\sqrt{2}}(5^x-1)[{\log }_{\sqrt{2}}\left(2\sqrt{2}\right)-{\log }_{\sqrt{2}}(5^x-1)]>2$

${\log }_{\sqrt{2}}(5^x-1)[3-{\log }_{\sqrt{2}}(5^x-1)]>2$

$3{\log }_{\sqrt{2}}(5^x-1)-{\left[{\log }_{\sqrt{2}}(5^x-1)\right]}^2>2$

${\left[{\log }_{\sqrt{2}}(5^x-1)\right]}^2-3{\log }_{\sqrt{2}}(5^x-1)+2<0$

$({\log }_{\sqrt{2}}(5^x-1)-2)({\log }_{\sqrt{2}}(5^x-1)-1)<0$

${\log }_{\sqrt{2}}(5^x-1)ϵ(1,2)$

$(5^x-1)ϵ(\sqrt{2},2)$

$5^xϵ(\sqrt{2}+1,3)$

$xϵ({\log }_5(\sqrt{2}+1),{\log }_{5}3)$

Also for, $\log (5^x-1)$

$5^x-1>0$

$5^x>1$

$x>0$

Taking intersection we get $xϵ({\log }_5(\sqrt{2}+1),{\log }_{5}3)$