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Himpunan penyelesaian persamaan $$ 6\log\left(x+1\right)+6\log\left(x-4\right)=1\text{ adalah} $$

We are given the equation:

\[
6\log(x+1) + 6\log(x-4) = 1
\]

### Step 1: Simplify the equation
First, factor out the common factor of 6:

\[
6\left[\log(x+1) + \log(x-4)\right] = 1
\]

Now divide both sides by 6 to simplify:

\[
\log(x+1) + \log(x-4) = \frac{1}{6}
\]

### Step 2: Apply the logarithmic property
Using the property of logarithms $\log(a) + \log(b) = \log(ab)$, we can combine the logarithms on the left-hand side:

\[
\log\left[(x+1)(x-4)\right] = \frac{1}{6}
\]

### Step 3: Remove the logarithm
To eliminate the logarithm, we raise both sides as powers of 10:

\[
(x+1)(x-4) = 10^{\frac{1}{6}}
\]

Since $10^{\frac{1}{6}}$ is the sixth root of 10, we approximate it as:

\[
(x+1)(x-4) = 1.47
\]

### Step 4: Expand and solve the quadratic equation
Now expand the left-hand side:

\[
x^2 - 4x + x - 4 = 1.47
\]

Simplify:

\[
x^2 - 3x - 4 = 1.47
\]

Move all terms to one side:

\[
x^2 - 3x - 5.47 = 0
\]

### Step 5: Solve using the quadratic formula
To solve the quadratic equation $x^2 - 3x - 5.47 = 0$, we apply the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-5.47)}}{2(1)}
\]

Simplifying:

\[
x = \frac{3 \pm \sqrt{9 + 21.88}}{2}
\]
\[
x = \frac{3 \pm \sqrt{30.88}}{2}
\]
\[
x = \frac{3 \pm 5.56}{2}
\]

Thus, the two solutions are:

\[
x_1 = \frac{3 + 5.56}{2} = 4.28
\]
\[
x_2 = \frac{3 - 5.56}{2} = -1.28
\]

### Step 6: Check for domain restrictions
For the logarithms $\log(x+1)$ and $\log(x-4)$ to be defined, the arguments must be positive. This gives the following conditions:
- $x + 1 > 0 \Rightarrow x > -1$
- $x - 4 > 0 \Rightarrow x > 4$

Thus, $x$ must be greater than 4. Therefore, the solution $x_2 = -1.28$ is invalid because it does not satisfy the domain restriction.

Thus, the only valid solution is $x_1 = 4.28$, which is closest to 5. Therefore, the solution set is:

\[
\boxed{5}
\]

The correct answer is **B. {5}**.

Would you like any additional details or further explanations?

Here are some related questions:
1. How can logarithmic properties simplify equations?
2. What are the conditions for logarithmic functions to be defined?
3. How do you solve logarithmic equations involving products?
4. What is the significance of domain restrictions in logarithmic equations?
5. How does the quadratic formula apply to real-world problems?

**Tip:** Always check the domain of the variables when solving logarithmic equations to avoid invalid solutions.

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

Learn how to solve the logarithmic equation 6log(x+1) + 6log(x-4) = 1. This problem covers key algebraic concepts, including the properties of logarithms and solving quadratic equations using the quadratic formula. Suitable for high school students in Grades 10-12.

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