Start by dividing both sides of the equation by 2:

\(\ln(5 - e^{-2x}) = \frac{1}{2}\).

Remove the logarithm by exponentiating both sides:

\(5 - e^{-2x} = e^{\frac{1}{2}}\).

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

\(e^{-2x} = 5 - e^{\frac{1}{2}}\).

Take the natural logarithm of both sides:

\(-2x = \ln(5 - e^{\frac{1}{2}})\).

Solve for \(x\):

\(x = -\frac{1}{2} \ln(5 - e^{\frac{1}{2}})\).

Calculate the value to 3 significant figures:

\(x \approx -0.605\).