Question

Simplify the following using the laws of logarithms:

(a) \(\quad \log _9 27\)
(b) \(\quad \log _3 2 \cdot \log _2 5 \cdot \log _5 9\)

Answer 1

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