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

Use the properties of logarithms to combine the logarithms: \(\ln(x+2) - \ln x = 2\).

This simplifies to \(\ln\left(\frac{x+2}{x}\right) = 2\).

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

Multiply both sides by \(x\) to get \(x+2 = e^2 x\).

Rearrange to solve for \(x\): \(x - e^2 x = -2\).

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

Solve for \(x\): \(x = \frac{-2}{1 - e^2}\).

Calculate \(x\) to three decimal places: \(x \approx 0.313\).