The number of solutions $log (3 x^{2} + x - 2) = 3 log (3 x - 2)$ is:

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Solution

$log (3 x^{2} + x - 2) = 3 log (3 x - 2)$
$3 x^{2} + x - 2 = {(3 x - 2)}^{3}$
$3 x^{2} + x - 2 = {(3 x)}^{3} - {(2)}^{3} - 3 (3 x) (2) (3 x - 2)$
$3 x^{2} + x - 2 = 27 x^{3} - 8 - 18 x (3 x - 2)$
$3 x^{2} + x - 2 = 27 x^{3} - 8 - 54 x^{2} + 36 x$
$27 x^{3} - 57 x^{2} + 35 x - 6 = 0$
$t h e r e f o r e 3 s o l u t i o n s a r e p o s s i b l e$
$b u t a s x \neq \frac{2}{3}$
$t h e r e f o r e o n l y t w o s o l u t i o n s o f x a r e p o s s i b l e .$

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