https://file.aiquickdraw.com/mathbot/file/17289641339193sualm8r.jpg

The given equation is:

\[
2 \log_3(x) + \log_3(a) = 0
\]

### Step-by-step solution:

1. **Isolate one logarithmic term**:
   Start by moving \(\log_3(a)\) to the other side of the equation:
   \[
   2 \log_3(x) = -\log_3(a)
   \]

2. **Simplify the logarithmic term**:
   Use the logarithmic property \(n \log_b(m) = \log_b(m^n)\) to rewrite \(2 \log_3(x)\):
   \[
   \log_3(x^2) = -\log_3(a)
   \]

3. **Combine the logs**:
   Now, apply the rule that \(-\log_b(a) = \log_b(1/a)\):
   \[
   \log_3(x^2) = \log_3\left(\frac{1}{a}\right)
   \]

4. **Set the arguments equal**:
   Since the logarithms on both sides have the same base, we can equate their arguments:
   \[
   x^2 = \frac{1}{a}
   \]

5. **Solve for \(x\)**:
   Take the square root of both sides to solve for \(x\):
   \[
   x = \pm \frac{1}{\sqrt{a}}
   \]

Thus, the solution is:
\[
x = \pm \frac{1}{\sqrt{a}}
\]

Would you like further details on any of the steps, or have any other questions?

### Related Questions:
1. What are the properties of logarithms that were used in this solution?
2. Can \(a\) be any value, or are there restrictions on \(a\)?
3. How would the solution change if the base of the logarithms were different?
4. Can you solve this equation graphically? What would that process look like?
5. How does this equation relate to exponential functions?

### Tip:
Always check that the values of \(x\) and \(a\) satisfy the domain restrictions of the logarithmic functions (e.g., \(x > 0\) and \(a > 0\)).

Step-by-step solution for solving the logarithmic equation 2 log_3(x) + log_3(a) = 0. Understand how to apply logarithmic properties and algebraic techniques to find the value of x. Suitable for Grades 10-12.

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