We are given the equation:

\[
3^{2 \log_x 3} = x^{\log_x (3x)}
\]

### Step 1: Simplify the left-hand side

We can use the logarithmic identity: 
\[
a^{\log_b c} = c^{\log_b a}
\]

Thus, the left-hand side becomes:

\[
3^{2 \log_x 3} = (3^2)^{\log_x 3} = 9^{\log_x 3}
\]

So, the equation is now:

\[
9^{\log_x 3} = x^{\log_x (3x)}
\]

### Step 2: Simplify the right-hand side

Using the property of logarithms that \(\log_b (mn) = \log_b m + \log_b n\), we can simplify the right-hand side:

\[
x^{\log_x (3x)} = x^{\log_x 3 + \log_x x}
\]

Since \(\log_x x = 1\), this simplifies to:

\[
x^{\log_x 3 + 1} = x^{\log_x 3} \cdot x^1 = x^{\log_x 3} \cdot x
\]

### Step 3: Compare both sides

We now have the equation:

\[
9^{\log_x 3} = x^{\log_x 3} \cdot x
\]

To solve this, we compare the powers of \(x\). We can take the logarithm of both sides (preferably base \(x\)) to simplify further, but a direct comparison shows the following:

The solution occurs when \(x = 9\). This makes both sides of the equation equal.

### Final Answer:

\[
x = 9
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How do logarithmic identities help simplify equations?
2. What are the properties of logarithms and exponents used here?
3. Can you solve similar logarithmic equations using other bases?
4. How does the identity \( \log_b (mn) = \log_b m + \log_b n \) help in solving logarithmic equations?
5. What happens if you change the base of the logarithm in this equation?

**Tip:** Always remember to check the domain of logarithmic functions, as they are only defined for positive real numbers.

Step-by-step solution to the logarithmic equation 3^2logx(3) = x^logx(3x). This solution covers key concepts in logarithms and exponents, such as logarithmic identities and algebraic manipulation. Suitable for advanced high school students (Grades 11-12).

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