if log3(7) = x and log2(9) = y determine an expression for log 9(7) + log2(3) in terms of x and y

We are given:

\[
\log_3(7) = x \quad \text{and} \quad \log_2(9) = y
\]

We are tasked with expressing \( \log_9(7) + \log_2(3) \) in terms of \( x \) and \( y \).

### Step 1: Express \( \log_9(7) \) in terms of \( \log_3(7) \)

We know that:

\[
\log_9(7) = \frac{\log_3(7)}{\log_3(9)}
\]

Since \( 9 = 3^2 \), we can simplify \( \log_3(9) \):

\[
\log_3(9) = \log_3(3^2) = 2
\]

Thus:

\[
\log_9(7) = \frac{\log_3(7)}{2} = \frac{x}{2}
\]

### Step 2: Express \( \log_2(3) \) in terms of \( \log_2(9) \)

Since \( 9 = 3^2 \), we can relate \( \log_2(9) \) and \( \log_2(3) \):

\[
\log_2(9) = 2 \log_2(3)
\]

Thus:

\[
y = 2 \log_2(3)
\]

Solving for \( \log_2(3) \), we get:

\[
\log_2(3) = \frac{y}{2}
\]

### Step 3: Combine the results

Now we can combine the expressions:

\[
\log_9(7) + \log_2(3) = \frac{x}{2} + \frac{y}{2}
\]

Finally, this simplifies to:

\[
\log_9(7) + \log_2(3) = \frac{x + y}{2}
\]

### Final Answer:

\[
\boxed{\frac{x + y}{2}}
\]

Would you like further details or have any questions?

Here are some related problems:
1. Solve for \( x \) and \( y \) if \( \log_3(7) = x \) and \( \log_2(9) = y \), and you are given \( \log_9(7) = 2 \).
2. Find \( \log_9(49) \) in terms of \( x \).
3. Express \( \log_3(14) \) in terms of \( x \) and \( \log_3(2) \).
4. Solve \( \log_2(27) \) if \( y = \log_2(9) \).
5. Express \( \log_3(21) \) in terms of \( x \) and \( \log_3(3) \).

**Tip**: When working with logarithmic identities, always look for opportunities to simplify by using exponent rules and logarithmic properties.

Learn how to express log9(7) + log2(3) in terms of x and y, given log3(7) = x and log2(9) = y. This problem explores logarithmic identities such as the change of base formula and logarithmic properties, suitable for students in Grades 10-12.

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