Start with the equation \(\ln(2 + e^{-x}) = 2\).

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

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

Take the natural logarithm of both sides: \(-x = \ln(e^2 - 2)\).

Multiply by -1 to solve for \(x\): \(x = -\ln(e^2 - 2)\).

Calculate the value: \(x \approx -1.68\).