Certainly! Let's go step-by-step to expand [tex]\(\log_9\left(\frac{2y-9}{y}\right)\)[/tex] using the quotient rule for logarithms.
### Step-by-Step Solution:
1. Identify the Quotient Rule:
The quotient rule for logarithms states that:
[tex]\[
\log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B)
\][/tex]
Here, [tex]\(A\)[/tex] is the numerator and [tex]\(B\)[/tex] is the denominator in the argument of the logarithm.
2. Apply the Quotient Rule:
In our problem, the expression inside the logarithm is [tex]\(\frac{2y-9}{y}\)[/tex]. So, we can identify [tex]\(A\)[/tex] as [tex]\(2y - 9\)[/tex] and [tex]\(B\)[/tex] as [tex]\(y\)[/tex].
According to the quotient rule:
[tex]\[
\log_9\left(\frac{2y-9}{y}\right) = \log_9(2y-9) - \log_9(y)
\][/tex]
3. Write the Expanded Form:
Now, substituting [tex]\(A\)[/tex] and [tex]\(B\)[/tex] into the quotient rule, we get:
[tex]\[
\log_9(2y-9) - \log_9(y)
\][/tex]
### Final Answer:
The expanded form of [tex]\(\log_9\left(\frac{2y-9}{y}\right)\)[/tex] using the quotient rule is:
[tex]\[
\log_9(2y-9) - \log_9(y)
\][/tex]
This completes the expansion using the quotient rule for logarithms.
