Start with the equation \(\ln(1 + x^2) = 1 + 2 \ln x\).

Use the properties of logarithms: \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\) and \(\ln a^b = b \ln a\).

Rearrange the equation: \(\ln(1 + x^2) - 2 \ln x = 1\).

This becomes \(\ln \left(\frac{1 + x^2}{x^2}\right) = 1\).

Exponentiate both sides to remove the logarithm: \(\frac{1 + x^2}{x^2} = e^1\).

Simplify: \(1 + x^2 = e x^2\).

Rearrange: \(1 = (e - 1)x^2\).

Thus, \(x^2 = \frac{1}{e - 1}\).

Take the square root: \(x = \sqrt{\frac{1}{e - 1}}\).

Calculate \(x\) using \(e \approx 2.718\):

\(x \approx \sqrt{\frac{1}{2.718 - 1}} \approx 0.763\).

Therefore, the solution is \(x = 0.763\) to 3 significant figures.