Question

The root(s) of the equation $({\log }_{3}x{)}^2-{\log }_{3}x=6$ is/are

• A. $\frac{1}{9}$
• B. $\sqrt{3}$
• C. $9$
• D. $27$

Answer 1

Answer: (A), (D)

The correct options are
A $\frac{1}{9}$
D $27$

$({\log }_{3}x{)}^2-{\log }_{3}x=6$
Let ${\log }_{3}x=t$
$\Rightarrow t^2-t-6=0$
$\Rightarrow t^2-3t+2t-6=0$
$\Rightarrow (t+2)(t-3)=0$
$\Rightarrow t=-2,3$
$\Rightarrow {\log }_{3}x=-2,3$
$\Rightarrow {\log }_{3}x=-2$ or ${\log }_{3}x=3$
$\Rightarrow x=3^{-2}$ or $x=3^3$
$\Rightarrow x=\frac{1}{9}$ or $x=27$