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Let's go through the expansion of each logarithmic expression one by one, using the properties of logarithms.

### Logarithmic Properties to Use
1. **Product Rule**: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
2. **Quotient Rule**: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
3. **Power Rule**: \(\log_b(x^k) = k \cdot \log_b(x)\)

### Expansions

1. **For \(\log_3(2a^2)\):**
   - Using the Product Rule: \(\log_3(2) + \log_3(a^2)\)
   - Then using the Power Rule on \(\log_3(a^2)\): \(\log_3(2) + 2\log_3(a)\)

2. **For \(\log_3(9a)\):**
   - Using the Product Rule: \(\log_3(9) + \log_3(a)\)
   - Since \(9 = 3^2\), we can simplify \(\log_3(9)\): \(2 + \log_3(a)\)

3. **For \(\log_3((3a)^2)\):**
   - Using the Power Rule: \(2 \cdot \log_3(3a)\)
   - Then applying the Product Rule within \(\log_3(3a)\): \(2(\log_3(3) + \log_3(a))\)
   - Simplify \(\log_3(3)\): \(2(1 + \log_3(a)) = 2 + 2\log_3(a)\)

4. **For \(\log_3\left(\frac{2}{a}\right)\):**
   - Using the Quotient Rule: \(\log_3(2) - \log_3(a)\)

5. **For \(\log_3\left(\frac{a}{2}\right)\):**
   - Using the Quotient Rule: \(\log_3(a) - \log_3(2)\)

### Matching Expanded Forms with Options

1. **\(\log_3(2a^2)\) → \(\log_3(2) + 2\log_3(a)\)**
2. **\(\log_3(9a)\) → \(2 + \log_3(a)\)**
3. **\(\log_3((3a)^2)\) → \(2 + 2\log_3(a)\)**
4. **\(\log_3\left(\frac{2}{a}\right)\) → \(\log_3(2) - \log_3(a)\)**
5. **\(\log_3\left(\frac{a}{2}\right)\) → \(\log_3(a) - \log_3(2)\)**

Let me know if you want further clarification or details on any step.

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Here are 5 related questions to expand your understanding:

1. How would you expand \(\log_b(x^3 y^2)\)?
2. How can you use the power rule to simplify \(\log_2(16)\)?
3. What is the difference between using the product and quotient rules in logarithmic expansions?
4. How would the expansion differ for \(\ln(xy)\) compared to \(\log(xy)\)?
5. Can logarithmic expressions with different bases be expanded and combined?

**Tip:** Always start with the power rule when you see an exponent inside the logarithm—it simplifies the expansion process significantly!

Expand each logarithmic expression on the left and match it with its expanded form on the right.

Learn how to expand logarithmic expressions by applying the product, quotient, and power rules of logarithms. This problem covers key algebraic concepts suitable for high school students and includes a matching exercise to reinforce understanding.

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