To convert the logarithmic equation [tex]\(\log _3 9 = x\)[/tex] to its exponential form, we begin by understanding what logarithms represent.
The expression [tex]\(\log_b(a) = c\)[/tex] translates to the statement [tex]\(b^c = a\)[/tex]. This means that the logarithm [tex]\(\log_b(a)\)[/tex] asks the question: "To what power must the base [tex]\(b\)[/tex] be raised, in order to produce [tex]\(a\)[/tex]?"
Given the specific logarithmic equation [tex]\(\log_3 9 = x\)[/tex], we can say that:
- The base [tex]\(b\)[/tex] is [tex]\(3\)[/tex].
- The result [tex]\(a\)[/tex] is [tex]\(9\)[/tex].
- The exponent [tex]\(c\)[/tex] that we're solving for is [tex]\(x\)[/tex].
So, we need to find the power [tex]\(x\)[/tex] such that [tex]\(3^x\)[/tex] equals [tex]\(9\)[/tex]. We can rewrite the logarithmic equation as:
[tex]\[ 3^x = 9 \][/tex]
Hence, the correct option is:
[tex]\[ 3^x = 9 \][/tex]
This translates the logarithmic expression into its equivalent exponential form.
The answer is the first choice:
[tex]\[ 3^x = 9 \][/tex]
