explain why there is only one solution, and discuss why extraneous solutions can exist

Let's break down why there is only one valid solution for the equation \(\log_5(x + 4) + \log_5(x - 4) = 2\), and why extraneous solutions can arise during the solving process.

### Reason for Only One Valid Solution:

1. **Logarithmic Domain Restrictions:**
   - The original equation involves the logarithmic expressions \(\log_5(x + 4)\) and \(\log_5(x - 4)\).
   - For a logarithm \(\log_b(y)\) to be defined, the argument \(y\) must be greater than 0.
   - Therefore, \(x + 4 > 0\) and \(x - 4 > 0\).
   - These inequalities imply \(x > 4\).

2. **Solving the Equation:**
   - The equation simplifies to \(x^2 - 16 = 25\), giving the solutions \(x = \pm\sqrt{41}\).
   - However, only the solution \(x = \sqrt{41}\) satisfies the condition \(x > 4\).
   - The other solution, \(x = -\sqrt{41}\), is invalid because it does not satisfy the domain restrictions. Specifically, \(x = -\sqrt{41}\) results in negative arguments for the logarithms, which are not defined in the real number system.

3. **Conclusion:**
   - Thus, there is only one valid solution: \(x = \sqrt{41}\), which meets the domain restriction.

### Why Extraneous Solutions Can Exist:

1. **Algebraic Manipulation:**
   - During the process of solving the equation, we combined the logarithmic terms into a single logarithm and then solved a quadratic equation. This step temporarily "ignores" the domain restrictions imposed by the logarithms.
   - The solution \(x = -\sqrt{41}\) is mathematically valid from the quadratic equation, but it does not meet the original logarithmic conditions (i.e., \(x > 4\)).

2. **Domain Considerations:**
   - Extraneous solutions arise when we solve equations without fully considering the domain restrictions imposed by the original equation's structure.
   - In this case, solving the quadratic equation \(x^2 - 16 = 25\) yields solutions that might not be valid for the logarithmic terms.

3. **Ensuring Validity:**
   - After finding potential solutions, we must always substitute them back into the original equation to ensure they satisfy all the conditions, particularly those related to the domain.
   - This step eliminates any extraneous solutions, ensuring that only the valid solution(s) remain.

### Summary:
- **Only one solution \(x = \sqrt{41}\) is valid** because it satisfies the domain restriction \(x > 4\).
- **Extraneous solutions** can occur when solving equations algebraically, especially when combining or simplifying terms, as these steps may generate solutions that do not meet the original conditions of the equation.

Would you like more details, or do you have further questions?

Here are some related questions you might find useful:
1. What steps should be taken to avoid extraneous solutions in logarithmic equations?
2. Can domain restrictions in other types of equations also lead to extraneous solutions?
3. How do you verify the validity of solutions in trigonometric equations?
4. What are common mistakes that lead to extraneous solutions in algebra?
5. Why is it important to check solutions in the context of the original equation?
6. How do you handle extraneous solutions in exponential equations?
7. What happens if you fail to consider domain restrictions in inequalities?
8. What are the key differences between extraneous solutions and invalid solutions?

**Tip:** Always be mindful of the domain when dealing with logarithmic and trigonometric equations, as these functions often impose restrictions that must be respected to avoid invalid solutions.

Explore why there is only one valid solution for the equation \(\log_5(x + 4) + \log_5(x - 4) = 2\) and the occurrence of extraneous solutions. Learn about logarithmic domain restrictions, solving quadratic equations, and ensuring solution validity. Suitable for high school students studying algebra and logarithms.

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