We are given the equation:

\[
\log_3 (x+1) = 1 + \log_3 (y)
\]

### Step-by-Step Solution:

1. **Rewrite the equation**:
   Start by isolating the logarithms on one side. Move the constant \(1\) to the left side:

   \[
   \log_3 (x+1) - \log_3 (y) = 1
   \]

2. **Apply the logarithmic subtraction rule**:
   Use the property of logarithms: \(\log_b A - \log_b B = \log_b \left( \frac{A}{B} \right)\):

   \[
   \log_3 \left( \frac{x+1}{y} \right) = 1
   \]

3. **Exponentiate both sides**:
   To eliminate the logarithm, exponentiate both sides using base 3:

   \[
   \frac{x+1}{y} = 3^1
   \]

   Simplifying the right-hand side:

   \[
   \frac{x+1}{y} = 3
   \]

4. **Solve for \(y\)**:
   Multiply both sides by \(y\):

   \[
   x+1 = 3y
   \]

5. **Solve for \(y\)**:

   \[
   y = \frac{x+1}{3}
   \]

### Final Answer:

\[
y = \frac{x+1}{3}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How do we derive the logarithmic properties used in this problem?
2. Can we solve the equation for \(x\) in terms of \(y\)?
3. What happens if the base of the logarithm changes, say to base 10?
4. How does the equation change if the constant 1 is replaced with another logarithmic term?
5. How can we verify this solution by substituting values of \(x\) and \(y\)?

**Tip**: When solving logarithmic equations, it's important to always check if the domain of the function (values of \(x\) or \(y\)) ensures that the arguments of the logarithms are positive.

Explore the solution to the logarithmic equation log_3(x+1) = 1 + log_3(y) with a step-by-step breakdown. Learn how logarithmic properties and exponentiation are applied to simplify and solve for y in terms of x. Ideal for students in Grades 9-12.

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