Use the properties of logarithms to condense the following expression as much as possible. writing the answer as a single term with a coefficient o All exponents should be positive. \[ \ln (4)+3 \ln (x)-3 \ln (y) \]

To condense the expression \(\ln (4) + 3 \ln (x) - 3 \ln (y)\) using the properties of logarithms, we can follow these steps: 1. **Use the power rule of logarithms**: The power rule states that \(a \ln(b) = \ln(b^a)\). We can apply this to \(3 \ln(x)\) and \(-3 \ln(y)\): - \(3 \ln(x) = \ln(x^3)\) - \(-3 \ln(y) = \ln(y^{-3})\) 2. **Rewrite the expression**: Substitute the results from step 1 into the original expression: \[ \ln(4) + \ln(x^3) + \ln(y^{-3}) \] 3. **Use the product rule of logarithms**: The product rule states that \(\ln(a) + \ln(b) = \ln(ab)\). We can combine the logarithms: \[ \ln(4) + \ln(x^3) = \ln(4x^3) \] Now, we can include \(\ln(y^{-3})\): \[ \ln(4x^3) + \ln(y^{-3}) = \ln(4x^3y^{-3}) \] 4. **Final expression**: The expression can be simplified to: \[ \ln\left(\frac{4x^3}{y^3}\right) \] Thus, the condensed expression is: \[ \ln\left(\frac{4x^3}{y^3}\right) \]