Start by rewriting the equation \(e^x + e^{2x} = e^{3x}\) as \(e^x + e^{2x} - e^{3x} = 0\).

Let \(u = e^x\), then the equation becomes \(u + u^2 - u^3 = 0\).

Factor the equation: \(u(u^2 + u - 1) = 0\).

Since \(u = e^x > 0\), we discard \(u = 0\) and solve \(u^2 + u - 1 = 0\).

Using the quadratic formula, \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = 1, c = -1\).

\(u = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2}\).

Choose the positive root: \(u = \frac{1 + \sqrt{5}}{2} \approx 1.618\).

Since \(u = e^x\), solve for \(x\): \(x = \ln\left(\frac{1 + \sqrt{5}}{2}\right)\).

Calculate \(x \approx 0.481\) to 3 significant figures.