To determine the value of [tex]\(\log_3 9\)[/tex], we need to find the exponent to which the base [tex]\(3\)[/tex] must be raised to generate the number [tex]\(9\)[/tex].
Let's express [tex]\(9\)[/tex] as a power of [tex]\(3\)[/tex]:
[tex]\[ 9 = 3^2 \][/tex]
Therefore,
[tex]\[ \log_3 9 = \log_3 (3^2) \][/tex]
Using the properties of logarithms, in particular the power rule which states [tex]\(\log_b (a^c) = c \log_b a\)[/tex], we have:
[tex]\[ \log_3 (3^2) = 2 \log_3 3 \][/tex]
Since [tex]\(\log_3 3 = 1\)[/tex] (because any log of a number to its own base is 1), we get:
[tex]\[ 2 \log_3 3 = 2 \cdot 1 = 2 \][/tex]
Hence,
[tex]\[ \log_3 9 = 2 \][/tex]
Therefore, the correct answer is:
C. 2
