Start with the equation:

\(\ln(2x - 1) = 2 \ln(x + 1) - \ln x\)

Apply the logarithm power rule: \(2 \ln(x + 1) = \ln((x + 1)^2)\)

So the equation becomes:

\(\ln(2x - 1) = \ln((x + 1)^2) - \ln x\)

Use the logarithm quotient rule: \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\)

\(\ln(2x - 1) = \ln\left(\frac{(x + 1)^2}{x}\right)\)

Since the logarithms are equal, the arguments must be equal:

\(2x - 1 = \frac{(x + 1)^2}{x}\)

Multiply both sides by \(x\) to clear the fraction:

\(x(2x - 1) = (x + 1)^2\)

Expand both sides:

\(2x^2 - x = x^2 + 2x + 1\)

Rearrange to form a quadratic equation:

\(2x^2 - x - x^2 - 2x - 1 = 0\)

\(x^2 - 3x - 1 = 0\)

Solve the quadratic equation using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a = 1\), \(b = -3\), \(c = -1\)

\(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}\)

\(x = \frac{3 \pm \sqrt{9 + 4}}{2}\)

\(x = \frac{3 \pm \sqrt{13}}{2}\)

Calculate the positive root:

\(x = \frac{3 + \sqrt{13}}{2} \approx 3.303\)