First, find the zeros of the function g. To do so, set g(x)=0 and solve for x.
[tex]\begin{gathered} g(x)=\log _5|2\log _3x| \\ g(x)=0 \\ \Rightarrow\log _5|2\log _3x|=0 \\ \Rightarrow|2\log _3x|=5^0 \\ \Rightarrow|2\log _3x|=1 \end{gathered}[/tex]
To solve the equation involving the absolute value, consider two cases.
Case 1. If the expression inside the absolute value is positive, then:
[tex]\begin{gathered} |2\log _3x|=2\log _3x \\ \Rightarrow2\log _3x=1 \\ \Rightarrow\log _3x=\frac{1}{2} \\ \Rightarrow x=3^{\frac{1}{2}} \end{gathered}[/tex]
Case 2. If the expression inside the absolute value is negative, then:
[tex]\begin{gathered} |2\log _3x|=1 \\ \Rightarrow-2\log _3x=1 \\ \Rightarrow\log _3x=-\frac{1}{2} \\ \Rightarrow x=3^{-\frac{1}{2}} \end{gathered}[/tex]
Then, the zeros of the function g are x=3^(1/2) and x=3^(-1/2).
Find the product of the zeros of g by multiplying them:
[tex]3^{\frac{1}{2}}\times3^{-\frac{1}{2}}=3^{\frac{1}{2}-\frac{1}{2}}=3^0=1[/tex]
Therefore, the product of the zeros of g is 1.
