Question

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

• A. Exactly three roots (real)
• B. At least one real root
• C. Exactly one irrational root
• D. All of these

Answer 1

Answer: (D)

Given equation is $x^{{\left(lo{g}_{3}x\right)}^2-\frac{9}{2}\left(lo{g}_{3}x\right)+5}=3^{3/2}$
let $lo{g}_{3}x=y$
$\Rightarrow x=3^y$
$\therefore x^{y^2-\frac{9}{2}y+5}=3^{3/2}$
$\therefore \left(y^2-\frac{9}{2}y+5\right)lo{g}_{3}x=\frac{3}{2}lo{g}_{3}3$
(taking log on both sides on base 3)
or $\left(y^2-\frac{9}{2}y+5\right)y=\frac{3}{2}$
or, $2y^3-9y^2+10y-3=0$
or $\left(y-1\right)\left(y-3\right)\left(2y-1\right)=0$
or, $y=1$ or $y=3$ or $y=\frac{1}{2}$
Now, $x=3^y$
$\Rightarrow x=3^1.3^3.3^{12}$ in which all three are real and one is irrational