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

Use the logarithm power rule: \(2 \ln x = \ln(x^2)\).

Substitute to get: \(\ln(x^2 + 1) = 1 + \ln(x^2)\).

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

Use the logarithm quotient rule: \(\ln\left(\frac{x^2 + 1}{x^2}\right) = 1\).

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

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

Rearrange to find \(x^2\): \(\frac{1}{x^2} = e - 1\).

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

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

Calculate \(x\) to 3 significant figures: \(x \approx 0.763\).