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.