Question

Solve the following equation:
$2({\log }_x\sqrt{5}{)}^2-3{\log }_x\sqrt{5}+1=0$

Answer 1

$2({\log }_x\sqrt{5}{)}^2-3{\log }_x\sqrt{5}+1=0let{\log }_x\sqrt{5}=a{2a}^2-3a+1=0\Rightarrow {2a}^2-a-2a+1=0\Rightarrow (a-1\left)\right(2a-1)=0\Rightarrow a=1,\frac{1}{2}\Rightarrow {\log }_x\sqrt{5}=1,{\log }_x\sqrt{5}=\frac{1}{2}\Rightarrow x=\sqrt{5},x={\left(\sqrt{5}\right)}^2=5\Rightarrow x=\sqrt{5},5.$