To solve the problem, we need to convert the given logarithmic equation into its equivalent exponential form. The logarithmic equation given is:
[tex]\[
\log _3 9 = 2
\][/tex]
To interpret this equation, recall that a logarithm answers the question: "To what power must the base (in this case, 3) be raised to produce a given number (in this case, 9)?"
This equation can be written as:
[tex]\[
\log _b a = c \quad \text{is equivalent to} \quad b^c = a
\][/tex]
Here, [tex]\( b \)[/tex] is the base, [tex]\( a \)[/tex] is the number, and [tex]\( c \)[/tex] is the exponent. For our specific problem, we identify:
- [tex]\( b = 3 \)[/tex] (the base)
- [tex]\( a = 9 \)[/tex] (the number)
- [tex]\( c = 2 \)[/tex] (the exponent)
Plugging these values into the exponential form, we get:
[tex]\[
3^2 = 9
\][/tex]
Hence, the equation [tex]\( \log _3 9 = 2 \)[/tex] is equivalent to:
[tex]\[
3^2 = 9
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{3^2 = 9}
\][/tex]
This is option A. Therefore, option A is the correct answer.
