Write the following expression as a sum and/or difference of logarithms. Express powers as factors. \( \ln \frac{x-3}{e^{8 x}} \) \( \ln \frac{x-3}{e^{8 x}}=\square \) (Simplify your answer.)

To express the given logarithmic expression \( \ln \frac{x-3}{e^{8 x}} \) as a sum and/or difference of logarithms, we can use the properties of logarithms. Specifically, we will use the following properties: 1. \( \ln \frac{a}{b} = \ln a - \ln b \) 2. \( \ln a^b = b \ln a \) Now, let's apply these properties step by step: 1. Start with the expression: \[ \ln \frac{x-3}{e^{8 x}} \] 2. Apply the quotient rule for logarithms: \[ \ln \frac{x-3}{e^{8 x}} = \ln(x-3) - \ln(e^{8 x}) \] 3. Now, apply the power rule for logarithms to the second term: \[ \ln(e^{8 x}) = 8 x \ln e \] Since \( \ln e = 1 \), this simplifies to: \[ \ln(e^{8 x}) = 8 x \] 4. Substitute this back into the expression: \[ \ln(x-3) - 8 x \] Thus, the final expression is: \[ \ln \frac{x-3}{e^{8 x}} = \ln(x-3) - 8 x \] So, the simplified answer is: \[ \ln(x-3) - 8 x \]