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

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**(a)**  
We start with the expression:  
$$
3 \ln e^{2}+\ln e^{-12}
$$  
Using the logarithm property $$\ln(e^x)=x$$, we have:  
$$
3 \ln e^{2}=3\cdot 2=6 \quad \text{and} \quad \ln e^{-12}=-12.
$$  
Thus,  
$$
3 \ln e^{2}+\ln e^{-12}=6+(-12)=-6.
$$

**(b)**  
We start with the expression:  
$$
\log_{5}75-\log_{5}3.
$$  
Using the logarithm property $$\log_{b}M-\log_{b}N=\log_{b}\left(\frac{M}{N}\right)$$, we can combine the logs:  
$$
\log_{5}75-\log_{5}3=\log_{5}\left(\frac{75}{3}\right)=\log_{5}25.
$$  
Since $$25=5^{2}$$, we have:  
$$
\log_{5}25=\log_{5}5^{2}=2.
$$

Thus, the answers are:  
$$
\text{(a)}\ -6 \quad \text{and} \quad \text{(b)}\ 2.
$$
**(a)** $$ 3 \ln e^{2}+\ln e^{-12} = -6 $$ **(b)** $$ \log_{5}75 - \log_{5}3 = 2 $$

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

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