The given equation is:

$\log_{1+x}(x-4) = \log_{x-4}(1+x)$

To solve this, follow these steps:

Step 1: Change of Base Formula

Using the change of base formula: $\log_a(b) = \frac{\log(b)}{\log(a)},$ we rewrite the equation:

$\frac{\log(x-4)}{\log(1+x)} = \frac{\log(1+x)}{\log(x-4)}.$

Step 2: Cross-Multiply

Cross-multiplying yields: $\log(x-4) \cdot \log(x-4) = \log(1+x) \cdot \log(1+x),$ or equivalently: $(\log(x-4))^2 = (\log(1+x))^2.$

Step 3: Take the Square Root

Taking the square root on both sides: $\log(x-4) = \pm \log(1+x).$

This gives two cases:

1. $\log(x-4) = \log(1+x)$,
2. $\log(x-4) = -\log(1+x)$.

Case 1: $\log(x-4) = \log(1+x)$

If the logarithms are equal: $x-4 = 1+x.$ Simplify: $-4 = 1,$ which is a contradiction. Thus, this case has no solution.

Case 2: $\log(x-4) = -\log(1+x)$

Rewrite using properties of logarithms: $\log(x-4) + \log(1+x) = 0.$ Combine the logs: $\log((x-4)(1+x)) = 0.$ Exponentiating both sides (base 10): $(x-4)(1+x) = 1.$

Step 4: Solve the Quadratic Equation

Expand the left-hand side: $x^2 - 3x - 4 = 1.$ Simplify: $x^2 - 3x - 5 = 0.$

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 1$, $b = -3$, and $c = -5$: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-5)}}{2(1)}.$ $x = \frac{3 \pm \sqrt{9 + 20}}{2}.$ $x = \frac{3 \pm \sqrt{29}}{2}.$

Thus, the solutions are: $x = \frac{3 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{29}}{2}.$

Step 5: Verify Domain Restrictions

Since $\log(x-4)$ and $\log(1+x)$ are defined:

• $x - 4 > 0 \implies x > 4$,
• $1 + x > 0 \implies x > -1$.

Thus, the valid domain is $x > 4$. Only $x = \frac{3 + \sqrt{29}}{2}$ satisfies this.

Final Answer:

$x = \frac{3 + \sqrt{29}}{2}.$

Let me know if you want more details! Here are some follow-up questions:

1. What is the significance of the logarithm's domain in solving such equations?
2. How does the change of base formula help simplify logarithmic equations?
3. Can logarithmic equations have negative solutions?
4. What are the properties of logarithms used in solving this problem?
5. How do you verify solutions to logarithmic equations?

Tip: Always check the domain when solving equations involving logarithms to avoid extraneous solutions.