Question

Solve the following equation:
$(a^{{\log }_{b}x}{)}^2-5y^{{\log }_{b}x}+6=0$

Answer 1

$({}_a{\log }_{b}x{)}^2-5{}_a{\log }_{b}x+6=0\Rightarrow {(}_a{\log }_{b}x{)}^2-2{}_a{\log }_{b}x-3a({\log }_{b}x)+6=0\Rightarrow {}_a{\log }_{b}x=2,{}_a{\log }_{b}x=3\Rightarrow x={}_2{\log }_{a}b,x={}_3{\log }_{a}b\Rightarrow x={}_2{(\log }_{a}b),{}_3\left({\log }_{a}b\right)$