Solve the logarithmic equation.
\[
\log _{3} x=-2
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
(Simplify your answer. Type an integer or a fraction.)
B. The equation has no solution. The solution set is empty, $\varnothing$.
Answer
Final Answer: The solution to the logarithmic equation is \(x = \frac{1}{9}\). Therefore, the correct choice is A. The solution set is \(\boxed{\frac{1}{9}}\).
Step 1 ：The logarithmic equation is in the form \(\log_b a = n\), which can be rewritten in exponential form as \(b^n = a\). In this case, \(b = 3\), \(n = -2\), and \(a = x\). So, we can rewrite the equation as \(3^{-2} = x\).
Step 2 ：Substitute the values of \(b\) and \(n\) into the equation, we get \(x = 3^{-2}\).
Step 3 ：Solving for \(x\), we get \(x = \frac{1}{9}\).
Step 4 ：Final Answer: The solution to the logarithmic equation is \(x = \frac{1}{9}\). Therefore, the correct choice is A. The solution set is \(\boxed{\frac{1}{9}}\).
