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The equation given is:

\[
\log_3(3x + 4) - 7\log_9(x^2) + 6\log_3(x) = 2
\]

### Step-by-step solution:

1. **Change the base of the second term**:
   The logarithm \(\log_9(x^2)\) can be rewritten in terms of base 3 because \(9 = 3^2\). Using the change of base formula:

   \[
   \log_9(x^2) = \frac{\log_3(x^2)}{\log_3(9)} = \frac{\log_3(x^2)}{2}
   \]

   Thus, the second term becomes:

   \[
   7\log_9(x^2) = 7 \cdot \frac{\log_3(x^2)}{2} = \frac{7}{2} \cdot 2\log_3(x) = 7\log_3(x)
   \]

2. **Rewrite the equation**:
   Now, substitute this back into the equation:

   \[
   \log_3(3x + 4) - 7\log_3(x) + 6\log_3(x) = 2
   \]

   Simplifying the terms involving \(\log_3(x)\):

   \[
   \log_3(3x + 4) - \log_3(x) = 2
   \]

3. **Combine the logarithmic terms**:
   Using the logarithmic property \(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\), we can combine the logarithmic terms:

   \[
   \log_3\left(\frac{3x + 4}{x}\right) = 2
   \]

4. **Convert to exponential form**:
   Now, rewrite the logarithmic equation in its exponential form:

   \[
   \frac{3x + 4}{x} = 3^2
   \]

   Simplifying:

   \[
   \frac{3x + 4}{x} = 9
   \]

5. **Solve for \(x\)**:
   Multiply both sides by \(x\):

   \[
   3x + 4 = 9x
   \]

   Rearrange the equation:

   \[
   4 = 6x
   \]

   Solving for \(x\):

   \[
   x = \frac{4}{6} = \frac{2}{3}
   \]

Thus, the solution is \(x = \frac{2}{3}\).

Would you like further details or have any questions?

Here are five related questions:
1. How do you apply the change of base formula for logarithms?
2. What other properties of logarithms are useful for solving equations like this?
3. How can you convert between different bases of logarithms efficiently?
4. What are the steps to combine logarithms with different bases in one equation?
5. How do you check the domain restrictions when solving logarithmic equations?

**Tip**: Always check that the solution is valid by ensuring that the argument inside each logarithm remains positive for your solution.

Solve log_3(3x + 4) - 7log_9(x^2) + 6log_3(x) = 2

Explore the detailed step-by-step solution to the logarithmic equation log_3(3x + 4) - 7log_9(x^2) + 6log_3(x) = 2. This problem focuses on logarithmic properties, change of base, and solving for x. Suitable for high school students in Grades 10-12.

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