Use the properties of logarithms to CONDENSE the expression. \( 3 \ln 2+4 \ln x-\ln 5-2 \ln y= \)

To condense the expression \( 3 \ln 2 + 4 \ln x - \ln 5 - 2 \ln y \), we will use the properties of logarithms: 1. **Power Rule**: \( a \ln b = \ln(b^a) \) 2. **Product Rule**: \( \ln a + \ln b = \ln(ab) \) 3. **Quotient Rule**: \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \) Now, let's apply these rules step by step: 1. **Apply the Power Rule**: - \( 3 \ln 2 = \ln(2^3) = \ln(8) \) - \( 4 \ln x = \ln(x^4) \) - \( -2 \ln y = \ln(y^{-2}) \) So, we can rewrite the expression as: \[ \ln(8) + \ln(x^4) - \ln(5) + \ln(y^{-2}) \] 2. **Combine using the Product Rule**: - Combine \( \ln(8) + \ln(x^4) \) to get \( \ln(8x^4) \). - Now we have \( \ln(8x^4) - \ln(5) + \ln(y^{-2}) \). 3. **Apply the Quotient Rule**: - Combine \( \ln(8x^4) - \ln(5) \) to get \( \ln\left(\frac{8x^4}{5}\right) \). 4. **Combine with \( \ln(y^{-2}) \)**: - Now we have \( \ln\left(\frac{8x^4}{5}\right) + \ln(y^{-2}) \). - This can be combined using the Product Rule again: \[ \ln\left(\frac{8x^4}{5} \cdot y^{-2}\right) = \ln\left(\frac{8x^4}{5y^2}\right) \] Thus, the condensed expression is: \[ \ln\left(\frac{8x^4}{5y^2}\right) \]