Question

The equation $x^{({\log }_{3}x{)}^2-\frac{9}{2}{\log }_{3}x+5}=3\sqrt{3}$ has

• A. at least one real root
• B. exactly one real root
• C. exactly one irrational root
• D. complex roots

Answer 1

Answer: (A), (C)

The correct options are
A at least one real root
C exactly one irrational root

Let ${\log }_{3}x=t\Rightarrow x=3^t$
Then $(3^t{)}^{t^2-\frac{9}{2}t+5}=3\sqrt{3}$
$\Rightarrow 3^{t^3-\frac{9}{2}{t}^2+5t}=3^{\frac{3}{2}}$
$\Rightarrow t^3-\frac{9}{2}{t}^2+5t=\frac{3}{2}$
$\Rightarrow 2t^3-9t^2+10t-3=0$
$\Rightarrow (t-\frac{1}{2})(t-1)(t-3)=0$
$\Rightarrow t=\frac{1}{2},1,3$

$\therefore x=\sqrt{3},3,27$ are the roots of the given equation.