Problem

Use the properties of logarithms to evaluate each of the following expressions
(a) $\ln e^{8}-7 \ln e^{3}=$ П
(b) $\log _{6} 9+\log _{6} 4=\square$

Answer

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Answer

Final Answer: \(\ln e^{8}-7 \ln e^{3}=\) \(\boxed{-13}\) and \(\log _{6} 9+\log _{6} 4=\) \(\boxed{2}\)

Steps

Step 1 :Given expressions are \(\ln e^{8}-7 \ln e^{3}\) and \(\log _{6} 9+\log _{6} 4\)

Step 2 :Using the property of logarithms \(\ln e^{x} = x\), the first expression simplifies to \(8 - 7 \ln e^{3}\)

Step 3 :Using the property of logarithms \(a \log_b x = \log_b x^a\), the first expression further simplifies to \(8 - 7*3\)

Step 4 :Using the property of logarithms \(\log_b x + \log_b y = \log_b (x*y)\), the second expression simplifies to \(\log_6 36\)

Step 5 :Using the property of logarithms \(\log_b b^x = x\), the second expression further simplifies to 2

Step 6 :Final Answer: \(\ln e^{8}-7 \ln e^{3}=\) \(\boxed{-13}\) and \(\log _{6} 9+\log _{6} 4=\) \(\boxed{2}\)

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