https://file.aiquickdraw.com/mathbot/file/1729553077681yxztde4o.jpg

The equation provided in the image is:

\[
\log_9(x - 3) - \log_9 2 = \log_9 x
\]

### Step-by-step solution:

1. **Apply the logarithmic property**:
   \[
   \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)
   \]
   Using this on the left-hand side:
   \[
   \log_9 \left(\frac{x - 3}{2}\right) = \log_9 x
   \]

2. **Set the arguments equal**:
   Since both sides have \(\log_9\), we can set the arguments equal to each other:
   \[
   \frac{x - 3}{2} = x
   \]

3. **Solve for \(x\)**:
   Multiply both sides by 2 to get rid of the fraction:
   \[
   x - 3 = 2x
   \]
   Now, subtract \(x\) from both sides:
   \[
   -3 = x
   \]

Thus, the solution is:

\[
x = -3
\]

### Checking the solution:

We substitute \(x = -3\) into the original logarithmic equation:
\[
\log_9 (-3 - 3) - \log_9 2 = \log_9 (-3)
\]
This results in trying to take the logarithm of a negative number, which is undefined in the real number system. Therefore, **there is no solution** to this equation.

### Final Answer:
**No solution.**

---

Would you like further details or have any questions?

Here are some related questions to explore:

1. How do you solve logarithmic equations with different bases?
2. What happens when a logarithmic equation leads to a negative argument?
3. Can logarithms of negative numbers exist in any mathematical context (e.g., complex numbers)?
4. How do you simplify logarithmic expressions?
5. What are some real-world applications of logarithmic equations?

**Tip**: Always check for extraneous solutions when solving logarithmic equations, as some solutions may not be valid in the domain of the logarithmic function.

Solve for x: log_9(x - 3) - log_9 2 = log_9 x

Learn how to solve the logarithmic equation log_9(x - 3) - log_9 2 = log_9 x using properties of logarithms. The solution explores key algebraic concepts and checks for any extraneous solutions. Suitable for students in Grades 10-12.

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