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Himpunan penyelesaian persamaan ^ 6 log(x+1)+^ 6 log(x - 4) = 1 adalah

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The equation in the image is:

\[
\log_6(x + 1) + \log_6(x - 4) = 1
\]

We are tasked with finding the solution set for this equation.

### Solution Steps:

1. Use the logarithmic property:
\[
\log_b(a) + \log_b(c) = \log_b(a \cdot c)
\]
This simplifies the equation to:
\[
\log_6((x + 1)(x - 4)) = 1
\]

2. Exponentiate both sides to remove the logarithm:
\[
(x + 1)(x - 4) = 6^1
\]
Simplifying:
\[
(x + 1)(x - 4) = 6
\]

3. Expand and solve the quadratic equation:
\[
x^2 - 4x + x - 4 = 6
\]
\[
x^2 - 3x - 10 = 0
\]

4. Solve this quadratic equation using the quadratic formula:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 40}}{2}
\]
\[
x = \frac{3 \pm \sqrt{49}}{2}
\]
\[
x = \frac{3 \pm 7}{2}
\]

So, the solutions are:
\[
x = \frac{3 + 7}{2} = 5 \quad \text{or} \quad x = \frac{3 - 7}{2} = -2
\]

5. Check for domain restrictions:
- \( \log_6(x + 1) \) is defined when \( x + 1 > 0 \), so \( x > -1 \).
- \( \log_6(x - 4) \) is defined when \( x - 4 > 0 \), so \( x > 4 \).

Thus, \( x = 5 \) is the only valid solution.

### Final Answer:
The solution set is \( \{5\} \).

Would you like more details or have any questions? Here are some related questions for further practice:

1. What is the logarithmic property used to combine two logarithms?
2. How do you solve quadratic equations using the quadratic formula?
3. Why is it necessary to check for domain restrictions in logarithmic equations?
4. Can you solve \(\log_3(x - 2) + \log_3(x + 3) = 2\)?
5. What are some applications of logarithmic equations in real life?

**Tip:** Always check for domain restrictions when dealing with logarithmic and square root equations!

Himpunan penyelesaian persamaan ^6 log(x+1) + ^6 log(x - 4) = 1 adalah

This problem involves solving the logarithmic equation ^6 log(x+1) + ^6 log(x-4) = 1, covering logarithmic properties and solving a quadratic equation. Suitable for students in Grades 9-11.

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