Solve the equation $log ({3 x}^{2} + x - 2) = 3 log (3 x - 2)$
find the number of roots.

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Solution

This equation is equivalent to the system
${\begin{matrix} {3 x}^{2} + x - 2 > 0 {3 x}^{2} + x - 2 = {(3 x - 2)}^{3} \end{matrix}$
$\Rightarrow {\begin{matrix} (x - 2 / 3) (x - 2) > 0 (x - 2 / 3) ({9 x}^{2} + 13 x + 3) = 0 \end{matrix}$
$\Rightarrow {\begin{matrix} x < 2 / 3 a n d x > 2 x = \frac{2}{3}, x = \frac{13 \pm \sqrt{61}}{18} \end{matrix}$
Hence given eqaution has the only one root $x_{1} = \frac{13 + \sqrt{61}}{18}$

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