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To condense the given logarithmic expression, we'll use the following logarithmic rules:

1. **Product Rule:** \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\)
2. **Power Rule:** \(c \cdot \log_b(m) = \log_b(m^c)\)

The expression is:
\[
\log_3 6 + 2 \cdot \log_3 x + \frac{1}{2} \log_3 9
\]

### Step 1: Apply the Power Rule
1. \(2 \cdot \log_3 x = \log_3 (x^2)\)
2. \(\frac{1}{2} \log_3 9 = \log_3 (9^{\frac{1}{2}}) = \log_3 3\) (since \(9^{1/2} = 3\))

Now the expression becomes:
\[
\log_3 6 + \log_3 (x^2) + \log_3 3
\]

### Step 2: Apply the Product Rule
Combine all the logarithmic terms:
\[
\log_3 (6 \cdot x^2 \cdot 3)
\]

### Step 3: Simplify the Expression
Multiply inside the logarithm:
\[
\log_3 (18x^2)
\]

So, the condensed form of the logarithmic expression is:
\[
\boxed{\log_3 (18x^2)}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do we expand a single logarithm back into a sum of multiple logs?
2. What happens if we change the base of the logarithms?
3. Can logarithmic properties be applied to solve logarithmic equations?
4. How can we use the change of base formula to evaluate logs with different bases?
5. What are some real-world applications of logarithmic functions?

**Tip:** Remember, the power rule allows you to move coefficients inside a logarithm by turning them into exponents, which is useful for simplifying or condensing expressions.

Condense the logarithmic expression: log₃ 6 + 2·log₃ x + 1/2 log₃ 9

Learn how to condense the logarithmic expression log₃ 6 + 2·log₃ x + 1/2 log₃ 9 using logarithmic rules such as the product rule and power rule. This step-by-step solution is perfect for students in Grades 9-12 looking to master logarithmic functions and algebra.

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