The equation x^{(log _{3} x)^{2}-frac{9}{2} l | vedclass

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(A) Taking $\log _{3}$ on both sides, we get: $(\log _{3} x)^{2}-\frac{9}{2} \log _{3} x+5 = \log _{3} (3 \sqrt{3}) = \log _{3} (3^{3/2}) = \frac{3}{2

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at least one real root

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(A) Taking $\log _{3}$ on both sides, we get: $(\log _{3} x)^{2}-\frac{9}{2} \log _{3} x+5 = \log _{3} (3 \sqrt{3}) = \log _{3} (3^{3/2}) = \frac{3}{2}$. Let $t = \log _{3} x$. Then the equation becomes: $t^{2} - \frac{9}{2} t + 5 = \frac{3}{2}$. Multiplying by $2$, we get $2t^{2} - 9t + 10 = 3$, which simplifies to $2t^{2} - 9t + 7 = 0$. Factoring the quadratic: $(2t - 7)(t - 1) = 0$. Thus, $t = 1$ or $t = \frac{7}{2}$. For $t = 1$, $\log _{3} x = 1 \Rightarrow x = 3^{1} = 3$. For $t = \frac{7}{2}$, $\log _{3} x = \frac{7}{2} \Rightarrow x = 3^{7/2} = 27\sqrt{3}$. Both roots are real. Therefore, the equation has at least one real root.

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

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