Question

If $3^{({\log }_{9}x{)}^2-\frac{9}{2}{\log }_{9}x+5}=3\sqrt{3},$ then possible value(s) of $x$ is/are

• A. $9$
• B. $3^{7/2}$
• C. $3^7$
• D. $3^{5/2}$

Answer 1

Answer: (A), (C)

The correct options are
A $9$
C $3^7$

$3^{({\log }_{9}x{)}^2-\frac{9}{2}{\log }_{9}x+5}=3\sqrt{3}$
Taking $\log$ on both sides to the base $3,$ we get
$({\log }_{9}x{)}^2-\frac{9}{2}{\log }_{9}x+5=\frac{3}{2}$
Put $y={\log }_{9}x$
$y^2-\frac{9}{2}y+5=\frac{3}{2}\Rightarrow 2y^2-9y+7=0$
$\Rightarrow (2y-7)(y-1)=0$
$\Rightarrow y=\frac{7}{2},1$
$\Rightarrow {\log }_{9}x=1$ or ${\log }_{9}x=\frac{7}{2}$
$\Rightarrow x=9$ or $x=9^{7/2}=3^7$