The given equation is:

$\log_5(x + 4) = 1 - \log_5(x + 8)$

Step 1: Combine logarithmic terms

We begin by moving the logarithmic term $\log_5(x + 8)$ to the left-hand side:

$\log_5(x + 4) + \log_5(x + 8) = 1$

Step 2: Apply logarithmic property

Use the property of logarithms:

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

This simplifies the left-hand side:

$\log_5((x + 4)(x + 8)) = 1$

Step 3: Exponentiate both sides

To remove the logarithm, we exponentiate both sides using base 5:

$(x + 4)(x + 8) = 5^1$

$(x + 4)(x + 8) = 5$

Step 4: Expand and simplify

Now, expand the left-hand side:

$x^2 + 8x + 4x + 32 = 5$

$x^2 + 12x + 32 = 5$

Subtract 5 from both sides:

$x^2 + 12x + 27 = 0$

Step 5: Solve the quadratic equation

Now, solve the quadratic equation:

$x^2 + 12x + 27 = 0$

Using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = 12$, and $c = 27$. Substituting these values:

$x = \frac{-12 \pm \sqrt{12^2 - 4(1)(27)}}{2(1)}$

$x = \frac{-12 \pm \sqrt{144 - 108}}{2}$

$x = \frac{-12 \pm \sqrt{36}}{2}$

$x = \frac{-12 \pm 6}{2}$

Step 6: Find the solutions

We have two possible solutions:

$x_1 = \frac{-12 + 6}{2} = \frac{-6}{2} = -3$

$x_2 = \frac{-12 - 6}{2} = \frac{-18}{2} = -9$

Step 7: Check for extraneous solutions

For logarithms to be valid, the arguments inside the logarithms must be positive. Therefore, we check the conditions:

• For $x = -3$, $\log_5(x + 4) = \log_5(1)$ and $\log_5(x + 8) = \log_5(5)$, both are valid.
• For $x = -9$, $\log_5(x + 4) = \log_5(-5)$, which is undefined (since logarithms of negative numbers are not allowed).

Thus, $x = -9$ is an extraneous solution.

Final Solution:

The only valid solution is:

$x = -3$

Would you like further clarification or more details on any step?

Here are 5 related questions:

1. How do you apply the properties of logarithms when combining terms?
2. What is the importance of checking for extraneous solutions in logarithmic equations?
3. How can you solve quadratic equations by factoring instead of using the quadratic formula?
4. What other methods can be used to solve logarithmic equations?
5. How does changing the base of a logarithm affect the equation?

Tip: Always check the domain of the logarithmic function to ensure the arguments are valid before finalizing your solution.