Start by using the properties of logarithms to combine the terms on the right-hand side:
\(\ln(2x - 3) = \ln\left(\frac{x^2}{x-1}\right)\).
Since the logarithms are equal, we equate the arguments:
\(2x - 3 = \frac{x^2}{x-1}\).
Multiply both sides by \(x-1\) to clear the fraction:
\((2x - 3)(x - 1) = x^2\).
Expand the left-hand side:
\(2x^2 - 2x - 3x + 3 = x^2\).
Simplify and combine like terms:
\(2x^2 - 5x + 3 = x^2\).
Subtract \(x^2\) from both sides:
\(x^2 - 5x + 3 = 0\).
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1, b = -5, c = 3\):
\(x = \frac{5 \pm \sqrt{25 - 12}}{2}\).
\(x = \frac{5 \pm \sqrt{13}}{2}\).
Calculate the roots:
\(x = \frac{5 + \sqrt{13}}{2} \approx 4.30\) and \(x = \frac{5 - \sqrt{13}}{2} \approx 0.70\).
Since \(x\) must be greater than 1 for the logarithms to be defined, the solution is:
\(x = 4.30\).
