Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible. In \( \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] \) \( \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right]=\square \)

1. Begin with the expression \[ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right]. \] 2. Use the logarithmic property for division: \[ \ln \left(\frac{A}{B}\right) = \ln(A) - \ln(B). \] Thus, \[ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = \ln\left(x^4 \sqrt{x^2+4}\right) - \ln\left((x+4)^4\right). \] 3. Use the logarithmic property for multiplication in the numerator: \[ \ln(AB) = \ln(A) + \ln(B). \] So, \[ \ln\left(x^4 \sqrt{x^2+4}\right) = \ln\left(x^4\right) + \ln\left(\sqrt{x^2+4}\right). \] 4. Apply the power rule for logarithms: \[ \ln\left(x^4\right) = 4\ln(x) \quad \text{and} \quad \ln\left((x+4)^4\right) = 4\ln(x+4). \] 5. Recall that a square root can be written as a power of one-half: \[ \sqrt{x^2+4} = (x^2+4)^{\frac{1}{2}}, \] which implies \[ \ln\left(\sqrt{x^2+4}\right) = \frac{1}{2}\ln\left(x^2+4\right). \] 6. Combine all of the results: \[ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = 4\ln(x) + \frac{1}{2}\ln\left(x^2+4\right) - 4\ln(x+4). \] Thus, the fully expanded logarithmic expression is \[ \boxed{4\ln(x) + \frac{1}{2}\ln\left(x^2+4\right) - 4\ln(x+4)}. \]