The problem involves simplifying the given logarithmic expression using logarithmic properties. Let's break down the expression step by step:

The expression is: $\frac{1}{2} \left( \log_3 x + \log_3 y \right) - 3 \log_3 (x+9)$

Step 1: Apply the logarithmic product rule

The product rule states that: $\log_b a + \log_b c = \log_b (a \cdot c)$ Using this, we can combine the first two terms: $\log_3 x + \log_3 y = \log_3 (x \cdot y)$

So the expression becomes: $\frac{1}{2} \log_3 (xy) - 3 \log_3 (x+9)$

Step 2: Apply the power rule

The power rule states that: $a \log_b x = \log_b (x^a)$ Now, apply this rule to both terms with constants: $\frac{1}{2} \log_3 (xy) = \log_3 ( (xy)^{1/2} )$ $-3 \log_3 (x+9) = \log_3 ( (x+9)^{-3} )$

Thus, the expression becomes: $\log_3 ( (xy)^{1/2} ) - \log_3 ( (x+9)^{-3} )$

Step 3: Apply the logarithmic quotient rule

The quotient rule states that: $\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)$ Using this rule, the expression becomes: $\log_3 \left( \frac{(xy)^{1/2}}{(x+9)^{-3}} \right)$

Step 4: Simplify the expression

The denominator has a negative exponent, which means we can move it to the numerator: $\log_3 \left( \frac{(xy)^{1/2}}{(x+9)^{-3}} \right) = \log_3 \left( (xy)^{1/2} \cdot (x+9)^3 \right)$

This is the simplified form of the logarithmic expression.

Step 5: Evaluate (if possible)

If we had specific values for $x$ and $y$, we could plug them in to simplify further or evaluate the logarithm. However, without specific values for $x$ and $y$, the expression is fully simplified as:

$\log_3 \left( (xy)^{1/2} \cdot (x+9)^3 \right)$

Let me know if you would like further clarification or have additional questions!

Here are five related questions for you to explore:

1. What is the product rule of logarithms used for in simplifying logarithmic expressions?
2. How does the power rule apply to logarithms, and what is its significance?
3. What does the quotient rule of logarithms allow us to do?
4. How can we evaluate logarithms with different bases using change of base formula?
5. Can we simplify logarithmic expressions with more complex exponents or multiple terms?

Tip: When working with logarithms, always apply the logarithmic properties (product, quotient, and power rules) to combine or separate terms as needed. This will make the simplification process much easier.