To find the value of [tex]\(\log_{27} 9\)[/tex], let's follow a step-by-step process.
1. Express the bases and arguments in terms of prime factors:
- We know that [tex]\(27\)[/tex] can be written as a power of [tex]\(3\)[/tex]: [tex]\(27 = 3^3\)[/tex].
- Similarly, [tex]\(9\)[/tex] can also be written as a power of [tex]\(3\)[/tex]: [tex]\(9 = 3^2\)[/tex].
2. Rewrite the logarithmic expression using these powers:
[tex]\[
\log_{27} 9 = \log_{3^3} 3^2
\][/tex]
3. Apply the change of base property for logarithms:
The change of base property states that [tex]\(\log_{a^m} a^n = \frac{n}{m}\)[/tex]. Here, [tex]\(a = 3\)[/tex], [tex]\(m = 3\)[/tex], and [tex]\(n = 2\)[/tex].
4. Substitute the values into the change of base formula:
[tex]\[
\log_{3^3} 3^2 = \frac{2}{3}
\][/tex]
Thus, the value of [tex]\(\log _{27} 9\)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
Therefore, the correct option is:
[tex]\[
\boxed{\frac{2}{3}}
\][/tex]
