https://r2.erweima.ai/mathbot/image/1726091338301uob8xtow.jpg

The given equation from the image is:

\[
\log_3 (x + 1) = \log_9 (10x + 1)
\]

### Steps to solve this equation:

1. First, recognize that \(\log_9\) can be rewritten using \(\log_3\) because \(9 = 3^2\). Using the logarithmic change of base formula:
   \[
   \log_9 (10x + 1) = \frac{1}{2} \log_3 (10x + 1)
   \]

2. The equation becomes:
   \[
   \log_3 (x + 1) = \frac{1}{2} \log_3 (10x + 1)
   \]

3. Multiply both sides by 2 to eliminate the fraction:
   \[
   2 \log_3 (x + 1) = \log_3 (10x + 1)
   \]

4. Using the logarithmic power rule \(\log_b (a^n) = n \log_b a\), we rewrite the left side:
   \[
   \log_3 ((x + 1)^2) = \log_3 (10x + 1)
   \]

5. Since the logarithms have the same base, we can set the arguments equal to each other:
   \[
   (x + 1)^2 = 10x + 1
   \]

6. Expand the left-hand side:
   \[
   x^2 + 2x + 1 = 10x + 1
   \]

7. Subtract \(10x + 1\) from both sides:
   \[
   x^2 + 2x + 1 - 10x - 1 = 0
   \]

8. Simplify the equation:
   \[
   x^2 - 8x = 0
   \]

9. Factor the quadratic equation:
   \[
   x(x - 8) = 0
   \]

10. Solve for \(x\):
    \[
    x = 0 \quad \text{or} \quad x = 8
    \]

### Check the solutions:
- For \(x = 0\): \(\log_3(0 + 1) = \log_9(10(0) + 1)\), which simplifies to \(\log_3(1) = \log_9(1)\). Both sides equal 0, so \(x = 0\) is a valid solution.
- For \(x = 8\): \(\log_3(8 + 1) = \log_9(10(8) + 1)\), simplifies to \(\log_3(9) = \log_9(81)\). Both sides equal 2, so \(x = 8\) is also a valid solution.

### Final Answer:
The solutions are \(x = 0\) and \(x = 8\).

Would you like more details or have any questions?

Here are five related questions:
1. What are the laws of logarithms used in this solution?
2. How would the solution change if the logarithmic bases were different?
3. What does the change of base formula state and how is it applied here?
4. Can you solve logarithmic equations with different bases without using the change of base formula?
5. What are the conditions under which logarithmic functions are defined?

**Tip:** Always ensure the arguments inside logarithms are positive for valid solutions!

Step-by-step solution for solving the logarithmic equation log₃(x + 1) = log₉(10x + 1). This problem involves logarithmic properties like change of base and the power rule, suitable for students in Grades 9-12.

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