First slide

Theory of equations

Question

The equation $x^{[{(\log_{3} x)}^{2} - (9 / 2) \log_{3} x + 5]} = 3 \sqrt{3}$ has

Moderate

A

at least one real solution
B

exactly three real solutions
C

exactly one irrational solution
D

complex roots

Solution

Given equation is
$\begin{aligned} x^{{\{\log_{3} x)}^{2} - (9 / 2) \log_{3} x + 5\}} = 3 \sqrt{3} \\ or x^{\{{(\log_{3} x)}^{2} - (9 / 2) \log_{3} x + 5\}} = 3^{3 / 2} \end{aligned}$
or $\{{(\log_{3} x)}^{2} - (9 / 2) \log_{3} x + 5\} \log_{3} x = \frac{3}{2} - - - - (i)$
Put log₃ X =t
Then from Eq. (i),
$\begin{array}{l} t^{3} - \frac{9}{2} t^{2} + 5 t = \frac{3}{2} \\ or 2 t^{3} - 9 t^{2} + 10 t - 3 = 0 \\ or (t - 1) (2 t^{2} - 7 t + 3) = 0 \\ or (t - 1) (t - \frac{1}{2}) (t - 3) = 0 \end{array}$
$\begin{array}{l} ∴ t = \frac{1}{2}, 1, 3 \\ or \log_{3} x = \frac{1}{2}, 1, 3 \\ \Rightarrow x = \sqrt{3}, 3, 27 . \end{array}$

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