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. \[ \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right) \]

To expand the expression \( \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right) \) using the properties of logarithms, we can follow these steps: 1. **Use the Quotient Rule**: The logarithm of a quotient can be expressed as the difference of the logarithms: \[ \ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b) \] 2. **Apply the Product Rule**: The logarithm of a product can be expressed as the sum of the logarithms: \[ \ln(ab) = \ln(a) + \ln(b) \] 3. **Simplify the Square Root**: The square root can be expressed as a power: \[ \sqrt{a} = a^{1/2} \] Now, let's apply these properties step by step to the given expression: 1. Start with the original expression: \[ \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right) \] 2. Apply the Quotient Rule: \[ \ln(x^{8} \sqrt{p^{7} q^{3}}) - \ln(e^{9}) \] 3. Simplify \( \ln(e^{9}) \): \[ \ln(e^{9}) = 9 \] 4. Now, simplify \( \sqrt{p^{7} q^{3}} \): \[ \sqrt{p^{7} q^{3}} = (p^{7} q^{3})^{1/2} = p^{7/2} q^{3/2} \] 5. Apply the Product Rule: \[ \ln(x^{8}) + \ln(p^{7/2}) + \ln(q^{3/2}) \] 6. Now, simplify each logarithm: - \( \ln(x^{8}) = 8 \ln(x) \) - \( \ln(p^{7/2}) = \frac{7}{2} \ln(p) \) - \( \ln(q^{3/2}) = \frac{3}{2} \ln(q) \) Putting it all together, we have: \[ 8 \ln(x) + \frac{7}{2} \ln(p) + \frac{3}{2} \ln(q) - 9 \] Thus, the expanded expression is: \[ 8 \ln(x) + \frac{7}{2} \ln(p) + \frac{3}{2} \ln(q) - 9 \]