Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated. without a calculator. \( \ln \left(\frac{x^{8} \sqrt{p^{1} q^{3}}}{e^{9}}\right) \)

1. We start with the expression \[ \ln \left(\frac{x^{8} \sqrt{p^{1} q^{3}}}{e^{9}}\right). \] 2. Apply the logarithm property \(\ln\frac{a}{b} = \ln a - \ln b\): \[ \ln \left(\frac{x^{8} \sqrt{p^{1} q^{3}}}{e^{9}}\right) = \ln\left(x^8 \sqrt{p\,q^3}\right) - \ln\left(e^9\right). \] 3. Expand \(\ln\left(x^8 \sqrt{p\,q^3}\right)\) using \(\ln(a b) = \ln a + \ln b\): \[ \ln\left(x^8 \sqrt{p\,q^3}\right) = \ln\left(x^8\right) + \ln\left(\sqrt{p\,q^3}\right). \] 4. Apply the power rule \(\ln\left(a^b\right) = b\ln(a)\) to each term: - For the first term: \[ \ln\left(x^8\right) = 8\ln(x). \] - For the second term, note that \(\sqrt{p\,q^3} = (p\,q^3)^{\frac{1}{2}}\), so: \[ \ln\left(\sqrt{p\,q^3}\right) = \frac{1}{2}\ln\left(p\,q^3\right). \] 5. Expand \(\ln\left(p\,q^3\right)\) using \(\ln(a b) = \ln a + \ln b\): \[ \ln\left(p\,q^3\right) = \ln(p) + \ln\left(q^3\right). \] Then, apply the power rule to \(\ln\left(q^3\right)\): \[ \ln\left(q^3\right) = 3\ln(q). \] So, \[ \ln\left(p\,q^3\right) = \ln(p) + 3\ln(q). \] 6. Substitute back: \[ \ln\left(\sqrt{p\,q^3}\right) = \frac{1}{2}\left[\ln(p) + 3\ln(q)\right] = \frac{1}{2}\ln(p) + \frac{3}{2}\ln(q). \] 7. Now, consider the term \(\ln\left(e^9\right)\). Using \(\ln\left(e^9\right) = 9\) (since \(\ln(e) = 1\)): \[ \ln\left(e^9\right) = 9. \] 8. Putting everything together: \[ \ln \left(\frac{x^{8} \sqrt{p\,q^{3}}}{e^{9}}\right) = 8\ln(x) + \frac{1}{2}\ln(p) + \frac{3}{2}\ln(q) - 9. \] Thus, the expanded expression is \[ 8\ln(x) + \frac{1}{2}\ln(p) + \frac{3}{2}\ln(q) - 9. \]