To evaluate [tex]\(\log_{12}(y^2)\)[/tex] given that [tex]\(\log_{12}(y) = 16\)[/tex], we can use the properties of logarithms.
One important property of logarithms is:
[tex]\[
\log_b(a^c) = c \cdot \log_b(a)
\][/tex]
In this case, we need to evaluate [tex]\(\log_{12}(y^2)\)[/tex]. According to the property stated above, we can rewrite [tex]\(\log_{12}(y^2)\)[/tex] as follows:
[tex]\[
\log_{12}(y^2) = 2 \cdot \log_{12}(y)
\][/tex]
We are given that [tex]\(\log_{12}(y) = 16\)[/tex]. Now we substitute [tex]\(\log_{12}(y) = 16\)[/tex] into our equation:
[tex]\[
\log_{12}(y^2) = 2 \cdot 16
\][/tex]
Simplifying this, we get:
[tex]\[
\log_{12}(y^2) = 32
\][/tex]
Therefore, the value of [tex]\(\log_{12}(y^2)\)[/tex] is [tex]\(\boxed{32}\)[/tex].
