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Simplify the following using the laws of logarithms:

(a) $\quad \log _9 27$
(b) $\quad \log _3 2 \cdot \log _2 5 \cdot \log _5 9$
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(a) There are two methods for simplifying this expression:

Method 1
Change to base 3
\begin{aligned} & \log _9 27 \\ & =\frac{\log _3 27}{\log _3 9} \\ & =\frac{\log _3 3^3}{\log _3 3^2} \\ & =\frac{3 \log _3 3}{2 \log _3 3} \\ & =\frac{3}{2} \end{aligned}

Method 2
Change to base 10
\begin{aligned} & \log _9 27 \\ & =\frac{\log _{10} 27}{\log _{10} 9} \\ & =\frac{\log 27}{\log 9} \\ & =\frac{\log 3^3}{\log 3^2} \\ & =\frac{3 \log 3}{2 \log 3}=\frac{3}{2} \end{aligned}

(b)
\begin{aligned} & \log _3 2 \cdot \log _2 5 \cdot \log _5 9 \\ & =\frac{\log 2}{\log 3} \cdot \frac{\log 5}{\log 2} \cdot \frac{\log 9}{\log 5} \\ & =\frac{\log 2}{\log 3} \cdot \frac{\log 5}{\log 2} \cdot \frac{\log 3^2}{\log 5} \\ & =\frac{\log 2}{\log 3} \cdot \frac{\log 5}{\log 2} \cdot \frac{2 \log 3}{\log 5} \\ & =2 \end{aligned}

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