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

Let's expand the logarithmic expression \( \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] \) using the properties of logarithms. First, we can use the property \( \ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b) \): \[ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = \ln(x^{4} \sqrt{x^{2}+4}) - \ln((x+4)^{4}). \] Next, we can break down \( \ln(x^{4} \sqrt{x^{2}+4}) \) further: \[ \ln(x^{4} \sqrt{x^{2}+4}) = \ln(x^{4}) + \ln(\sqrt{x^{2}+4}). \] Using the property \( \ln(a^{b}) = b \ln(a) \): \[ \ln(x^{4}) = 4\ln(x) \quad \text{and} \quad \ln(\sqrt{x^{2}+4}) = \frac{1}{2}\ln(x^{2}+4). \] Now, we handle the denominator: \[ \ln((x+4)^{4}) = 4\ln(x+4). \] Putting it all together, we have: \[ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = 4\ln(x) + \frac{1}{2}\ln(x^{2}+4) - 4\ln(x+4). \] So the final answer is: \[ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = 4 \ln(x) + \frac{1}{2} \ln(x^{2}+4) - 4 \ln(x+4). \]