https://file.aiquickdraw.com/mathbot/file/17290194026863lm8pk3d.jpg

The equation in the image is:

\[
\log_3{x} + \log_3{(x+2)} = 1
\]

### Step-by-step solution:

1. **Apply the logarithm property**: 
   The sum of two logarithms with the same base can be rewritten as the logarithm of the product:

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

   This simplifies to:

   \[
   \log_3[x(x+2)] = 1
   \]

2. **Exponentiate both sides**: 
   Since \(\log_3\) is the logarithm base 3, we can exponentiate both sides to remove the logarithm:

   \[
   3^{\log_3[x(x+2)]} = 3^1
   \]

   This simplifies to:

   \[
   x(x + 2) = 3
   \]

3. **Expand and simplify**: 
   Expand the left side:

   \[
   x^2 + 2x = 3
   \]

   Subtract 3 from both sides to set the equation equal to 0:

   \[
   x^2 + 2x - 3 = 0
   \]

4. **Factor the quadratic**: 
   Factor the quadratic equation:

   \[
   (x + 3)(x - 1) = 0
   \]

5. **Solve for \(x\)**: 
   Set each factor equal to 0:

   \[
   x + 3 = 0 \quad \text{or} \quad x - 1 = 0
   \]

   Solving these gives:

   \[
   x = -3 \quad \text{or} \quad x = 1
   \]

6. **Check for extraneous solutions**: 
   Logarithms are only defined for positive arguments, so we must check if these values of \(x\) are valid in the original equation.

   - For \(x = 1\), both \(\log_3{1}\) and \(\log_3{(1 + 2)}\) are defined.
   - For \(x = -3\), \(\log_3{(-3)}\) is not defined because logarithms of negative numbers are undefined in the real number system.

   Therefore, the only valid solution is \(x = 1\).

### Final Answer: 
The solution is \(x = 1\), which matches the selected answer.

---

Do you have any questions or would you like further details on any step?

Here are 5 related questions for practice:
1. How would the equation change if the logarithm base was 10 instead of 3?
2. Solve the equation \(\log_5{x} + \log_5{(x+4)} = 1\).
3. Can you solve \(\log_2{x} + \log_2{(x+3)} = 2\)?
4. What happens if we have \(\log_3{(x-2)} + \log_3{(x+5)} = 2\)?
5. What are the properties of logarithms that help simplify log expressions?

**Tip:** Always check if your logarithmic solutions make the arguments inside the log functions positive to avoid extraneous solutions.

Solve for the variable: log_3 x + log_3 (x+2) = 1

Learn how to solve the logarithmic equation log_3 x + log_3 (x+2) = 1. This problem covers important mathematical concepts like logarithmic properties, exponentiation, and quadratic equations. Suitable for students in Grades 9-12.

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