Use the properties of logarithms to CONDENSE the expression. \( 3 \ln 2+4 \ln x-\ln 5-2 \ln y= \)
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To condense the expression \( 3 \ln 2 + 4 \ln x - \ln 5 - 2 \ln y \), we can use the properties of logarithms. First, we can use the power property, which says \( a \ln b = \ln(b^a) \). Thus, we rewrite the terms: - \( 3 \ln 2 = \ln(2^3) = \ln 8 \) - \( 4 \ln x = \ln(x^4) \) - \( -\ln 5 = \ln\left(\frac{1}{5}\right) \) - \( -2 \ln y = \ln(y^{-2}) \) Now combining these using the addition and subtraction properties of logarithms: \[ \ln 8 + \ln(x^4) + \ln\left(\frac{1}{5}\right) + \ln(y^{-2}) = \ln\left(\frac{8 \cdot x^4 \cdot y^{-2}}{5}\right) \] So the condensed expression is: \[ \ln\left(\frac{8x^4}{5y^2}\right) \]
