Solve the equation
$2 log 2 x = log (7 x - 2 - {2 x}^{2})$

x = \frac{1}{2}

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x = \frac{2}{3}

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x = \frac{3}{2}

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x = \frac{5}{2}

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Solution

The correct options are
A $x = \frac{1}{2}$
D $x = \frac{2}{3}$
This equation is equivalent to the system
${\begin{matrix} 2 x > 0 {(2 x)}^{2} = 7 x - 2 - {2 x}^{2} \end{matrix}$
$\Rightarrow {\begin{matrix} x > 0 {6 x}^{2} - 7 x + 2 = 0 \end{matrix}$
$\Rightarrow ⎧ ⎨ ⎩ \begin{matrix} x > 0 (x - \frac{1}{2}) (x - \frac{2}{3}) = 0 \end{matrix}$
$\Rightarrow ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} x = \frac{1}{2} x = \frac{2}{3} \end{matrix}$
Hence, $x_{1} = \frac{1}{2}$ and $x_{2} = \frac{2}{3}$ are the roots of the original equation.

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