Start with the equation \(\ln(x^2 + 4) = 2 \ln x + \ln 4\).

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

Thus, the equation becomes \(\ln(x^2 + 4) = \ln(x^2) + \ln 4\).

Use the logarithm product rule: \(\ln(x^2) + \ln 4 = \ln(4x^2)\).

Now, the equation is \(\ln(x^2 + 4) = \ln(4x^2)\).

Since the logarithms are equal, set the arguments equal: \(x^2 + 4 = 4x^2\).

Rearrange to get: \(4x^2 - x^2 = 4\).

Simplify: \(3x^2 = 4\).

Divide by 3: \(x^2 = \frac{4}{3}\).

Take the square root: \(x = \frac{2}{\sqrt{3}}\).