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Q.
The equation $x^{\left(log_{3}\, x\right)^2-\frac{9}{2}log_{3} x+5} =3 \sqrt{3}$ has
Complex Numbers and Quadratic Equations
Solution:
Given equation is $x^{\left(log_{3}\, x\right)^2-\frac{9}{2}\left(log_{3} x\right)+5} =3^{3 /2}$
let $log_{3}x = y$
$\Rightarrow x = 3^{y}$
$\therefore x^{y^2 -\frac{9}{2}y+5}=3^{3 /2}$
$\therefore \left(y^{2}-\frac{9}{2}y+5\right)log_{3}\,x =\frac{3}{2}\,log_{3}\,3$
(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^{1 2} $ in which all three are real and one is irrational