Start with the equation:

\(\ln(2x^2 - 3) = 2 \ln x - \ln 2\)

Apply the logarithm laws:

\(2 \ln x - \ln 2 = \ln \left( \frac{x^2}{2} \right)\)

So the equation becomes:

\(\ln(2x^2 - 3) = \ln \left( \frac{x^2}{2} \right)\)

Remove the logarithms:

\(2x^2 - 3 = \frac{x^2}{2}\)

Multiply through by 2 to clear the fraction:

\(4x^2 - 6 = x^2\)

Rearrange the equation:

\(4x^2 - x^2 = 6\)

\(3x^2 = 6\)

Divide by 3:

\(x^2 = 2\)

Take the square root:

\(x = \sqrt{2}\) or \(x = -\sqrt{2}\)

Since \(x\) must be positive (as it is inside a logarithm), the solution is:

\(x = \sqrt{2}\)