To determine the value of [tex]\(\log_4 16\)[/tex], let's consider the definition of a logarithm. The logarithm [tex]\(\log_b a\)[/tex] answers the question: "To what power must [tex]\(b\)[/tex] be raised, to produce [tex]\(a\)[/tex]?"
In our case, we need to figure out what power [tex]\(4\)[/tex] must be raised to in order to get [tex]\(16\)[/tex]:
[tex]\[
\log_4 16 = x
\][/tex]
This can be rewritten in exponential form as:
[tex]\[
4^x = 16
\][/tex]
We know that [tex]\(16\)[/tex] can be written as a power of [tex]\(4\)[/tex]:
[tex]\[
16 = 4^2
\][/tex]
Thus, the equation becomes:
[tex]\[
4^x = 4^2
\][/tex]
Since the bases are the same, the exponents must be equal:
[tex]\[
x = 2
\][/tex]
Therefore, the value of [tex]\(\log_4 16\)[/tex] is:
[tex]\[
2
\][/tex]
Thus, the correct answer is:
[tex]\[
2
\][/tex]
