Use the properties of logarithms to evaluate each of the following expressions. \[ \begin{array}{l}\text { (a) } 3 \ln e^{2}+\ln e^{-12}=\square \\ \text { (b) } \log _{5} 75-\log _{5} 3=\square\end{array} \]

First, let's evaluate the expressions using the properties of logarithms. For (a): \[ 3 \ln e^{2} + \ln e^{-12} \] Using the property \( \ln e^{x} = x \): \[ 3 \cdot 2 + \ln e^{-12} = 6 + (-12) = 6 - 12 = -6 \] So for part (a): \[ 3 \ln e^{2} + \ln e^{-12} = -6 \] For (b): \[ \log_{5} 75 - \log_{5} 3 \] Using the property \( \log_{a} b - \log_{a} c = \log_{a} \left(\frac{b}{c}\right) \): \[ \log_{5} \left(\frac{75}{3}\right) = \log_{5} 25 \] And since \( 25 = 5^2 \): \[ \log_{5} 25 = 2 \] So for part (b): \[ \log_{5} 75 - \log_{5} 3 = 2 \] Final answers: (a) = -6, (b) = 2