To solve the equation:

\[
\log{(x+2)} + \log{(x-5)} = \log{4}
\]

### Step 1: Apply the logarithmic property
The logarithmic property \(\log{a} + \log{b} = \log{(a \cdot b)}\) allows us to combine the left side:

\[
\log{((x+2)(x-5))} = \log{4}
\]

### Step 2: Eliminate the logarithms
Since both sides of the equation are logarithms with the same base, we can drop the logs:

\[
(x+2)(x-5) = 4
\]

### Step 3: Expand and simplify
Now expand the left-hand side:

\[
x^2 - 5x + 2x - 10 = 4
\]

Simplifying the terms:

\[
x^2 - 3x - 10 = 4
\]

### Step 4: Set up the quadratic equation
Move all terms to one side:

\[
x^2 - 3x - 14 = 0
\]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-14)}}{2(1)}
\]

Simplifying:

\[
x = \frac{3 \pm \sqrt{9 + 56}}{2}
\]
\[
x = \frac{3 \pm \sqrt{65}}{2}
\]

Thus, the solutions are:

\[
x = \frac{3 + \sqrt{65}}{2} \quad \text{or} \quad x = \frac{3 - \sqrt{65}}{2}
\]

### Step 6: Check for extraneous solutions
Since the original equation involves logarithms, we must ensure the arguments of the logarithms are positive:
- For \(\log{(x+2)}\), we need \(x + 2 > 0 \Rightarrow x > -2\)
- For \(\log{(x-5)}\), we need \(x - 5 > 0 \Rightarrow x > 5\)

Thus, we only accept solutions where \(x > 5\). Therefore, we discard the solution \(\frac{3 - \sqrt{65}}{2}\) because it is less than 5.

### Final Answer:
The valid solution is:

\[
x = \frac{3 + \sqrt{65}}{2}
\]

Would you like more details or clarifications on any of these steps?

---

Here are five related questions for further practice:
1. Solve \(\log{(x+3)} + \log{(x-4)} = \log{7}\).
2. Solve \(\log{(x^2 - 4)} = \log{9}\).
3. What is the domain of the function \(f(x) = \log{(x+5)}\)?
4. Solve the equation \(\log_2{(x+1)} + \log_2{(x-2)} = 3\).
5. How would you solve \(\log_a{(x)} + \log_a{(y)} = \log_a{(z)}\) for \(x\)?

**Tip**: When solving logarithmic equations, always check the domain restrictions of the logarithmic functions to avoid extraneous solutions.

Learn how to solve the logarithmic equation log(x+2) + log(x-5) = log(4) by using properties of logarithms and converting it into a quadratic equation. This step-by-step solution helps you understand the concepts of combining logarithms and applying the quadratic formula. Suitable for high school students in Grades 10-12.

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