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.