Question

Solve the following equation:
${\log }_{x}2-{\log }_{4}x+\frac{7}{6}=0$

Answer 1

${\log }_{x}2-{\log }_{4}x+\frac{7}{6}=0\Rightarrow \frac{1}{{\log }_{2}x}-{\frac{1}{2}\log }_{2}x+\frac{7}{6}=0\Rightarrow 2-({{\log }_{2}x)}^2+\frac{7}{3}({\log }_{2}x)=0\Rightarrow 3({{\log }_{2}x)}^2-7\left({\log }_{2}x\right)-6=0\Rightarrow {\log }_{2}x=\frac{7\pm \sqrt{49+72}}{6}\Rightarrow {\log }_{2}x=3,\frac{-2}{3}\Rightarrow x=8,\sqrt[3]{3\frac{1}{4}}\Rightarrow x=8,2^{\frac{-2}{3}}$