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.

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!