Part 1: Rewrite the given equation without logarithms

We are given the equation:

$\log_2(x + 6) + \log_2(x + 5) = 1$

To rewrite this equation without logarithms, we can apply the logarithmic property:

$\log_b(A) + \log_b(B) = \log_b(A \cdot B)$

Using this property, we combine the two logarithms:

$\log_2\left[(x + 6)(x + 5)\right] = 1$

This is the rewritten equation without logarithms.

---

Part 2: Solve the equation

Now, we proceed to solve the equation. First, rewrite the equation as:

$\log_2\left[(x + 6)(x + 5)\right] = 1$

By the definition of logarithms, this is equivalent to:

$(x + 6)(x + 5) = 2^1$

Simplifying:

$(x + 6)(x + 5) = 2$

Now, expand the left side:

$x^2 + 5x + 6x + 30 = 2$

Simplify:

$x^2 + 11x + 30 = 2$

Subtract 2 from both sides:

$x^2 + 11x + 28 = 0$

Now, we factor the quadratic equation:

$(x + 7)(x + 4) = 0$

Setting each factor equal to 0 gives the potential solutions:

$x + 7 = 0 \quad \text{or} \quad x + 4 = 0$

$x = -7 \quad \text{or} \quad x = -4$

---

Domain Considerations

For logarithms to be valid, their arguments must be positive. In this case:

• $x + 6 > 0 \Rightarrow x > -6$
• $x + 5 > 0 \Rightarrow x > -5$

Therefore, the domain restriction is $x > -5$.

• $x = -7$ is not valid because it is less than -5.
• $x = -4$ is valid because it satisfies the domain restrictions.

Thus, the only valid solution is $x = -4$.

---

Final Answer

The solution set is:

$\boxed{x = -4}$

Do you have any questions or would you like further details?

Here are 5 related questions for further practice:

1. How do you solve logarithmic equations involving multiple logs with different bases?
2. What are the domain restrictions when solving logarithmic equations?
3. How can you apply the change of base formula to solve logarithmic equations?
4. What are some common mistakes made when solving logarithmic equations?
5. How do you check if a logarithmic solution is valid?

Tip: Always check the domain of logarithmic expressions to ensure your solutions are valid, as logarithms are only defined for positive values.