Questions: Use the properties of logarithms to condense the following expression as much as possible, writing the answer as a single term with 1. All exponents should be positive. 2(ln(sqrt[2](e^3)) - ln(xy))

Solution Steps

We start with the expression \( \ln \left(\sqrt[2]{\mathrm{e}^{3}}\right) \). Using the property of logarithms that states \( \ln(\sqrt{a}) = \frac{1}{2} \ln(a) \), we can rewrite this as: \[ \ln \left(\sqrt[2]{\mathrm{e}^{3}}\right) = \frac{1}{2} \ln(\mathrm{e}^{3}) = \frac{3}{2} \]

Next, we substitute this result back into the original expression: \[ 2\left(\frac{3}{2} - \ln(xy)\right) \]

Now, we distribute the \( 2 \) across the terms inside the parentheses: \[ 2 \cdot \frac{3}{2} - 2 \cdot \ln(xy) = 3 - 2\ln(xy) \]

The final condensed expression is: \[ 3 - 2\ln(xy) \]

Final Answer