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.