To solve the expression [tex]\(\frac{\log_9(m)}{\log(m)}\)[/tex], we can use the properties of logarithms. Let's go through the steps carefully.
1. Understanding the Change of Base Formula:
The change of base formula for logarithms states that:
[tex]\[
\log_b(a) = \frac{\log_k(a)}{\log_k(b)}
\][/tex]
where [tex]\(b\)[/tex] and [tex]\(a\)[/tex] are positive real numbers and [tex]\(k\)[/tex] is any positive real number different from 1.
2. Applying the Change of Base Formula:
We can rewrite [tex]\(\log_9(m)\)[/tex] using the change of base formula. Let’s choose the common logarithm (base 10) for simplicity:
[tex]\[
\log_9(m) = \frac{\log(m)}{\log(9)}
\][/tex]
3. Substitute [tex]\(\log_9(m)\)[/tex] into the Original Expression:
Now, substitute [tex]\(\log_9(m)\)[/tex] into the original expression [tex]\(\frac{\log_9(m)}{\log(m)}\)[/tex]:
[tex]\[
\frac{\log_9(m)}{\log(m)} = \frac{\frac{\log(m)}{\log(9)}}{\log(m)}
\][/tex]
4. Simplify the Expression:
Simplify the fraction:
[tex]\[
\frac{\frac{\log(m)}{\log(9)}}{\log(m)} = \frac{\log(m)}{\log(9) \cdot \log(m)}
\][/tex]
Since [tex]\(\log(m)\)[/tex] is in both the numerator and the denominator (given that [tex]\(\log(m) \neq 0\)[/tex]), we can cancel [tex]\(\log(m)\)[/tex]:
[tex]\[
\frac{1}{\log(9)}
\][/tex]
Therefore, the expression [tex]\(\frac{\log_9(m)}{\log(m)}\)[/tex] simplifies to [tex]\(\frac{1}{\log(9)}\)[/tex].
So, the correct choice is:
[tex]\[
\boxed{D \: \frac{1}{\log(9)}}
\][/tex]
