Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact answer. \\(\log_{6} x = 2\\)\
rewrite the given equation without logarithms. do not solve for x.\\(x = 6^{2}\\) (do not simplify.)\
solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\
a. the solution set is \\(\boxed{}\\) (simplify your answer.)\
b. there are infinitely many solutions.\
c. there is no solution.

Explanation:

Step1: Recall logarithm definition

The logarithmic equation \(\log_{b}a = c\) is equivalent to \(b^{c}=a\) (by the definition of logarithms, where \(b>0\), \(b
eq1\), and \(a>0\)).

Step2: Apply the definition to the given equation

For the equation \(\log_{6}x = 2\), we identify \(b = 6\), \(c = 2\), and \(a=x\). Using the definition \(b^{c}=a\), we substitute the values: \(6^{2}=x\).

Step3: Calculate \(6^{2}\)

We know that \(6^{2}=6\times6 = 36\). We also need to check the domain of the original logarithmic function \(\log_{6}x\). The domain of \(\log_{b}x\) is \(x>0\), and \(36>0\), so it is in the domain.

Answer:

The solution set is \(\{36\}\) (so the correct choice is A with \(x = 36\))