Solve the logarithmic equation. Be sure to reject any value of that is not in the domain of the original
logarithmic expressions. Give the exact answer.

Rewrite the given equation without logarithms. Do not solve for .

Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer
box to complete your choice.
A. The solution set is .
(Simplify your answer. Use a comma to separate answers as needed.)
O B. There are infinitely many solutions.
O C. There is no solution.

Ask by Ellis Ramirez. in the United States

Mar 31,2025

Question

Solve the logarithmic equation. Be sure to reject any value of that is not in the domain of the original
logarithmic expressions. Give the exact answer.

Rewrite the given equation without logarithms. Do not solve for .

Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer
box to complete your choice.
A. The solution set is .
(Simplify your answer. Use a comma to separate answers as needed.)
O B. There are infinitely many solutions.
O C. There is no solution.

Ask by Ellis Ramirez. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

**Step 1. Domain Restrictions**

The logarithmic functions are defined when their arguments are positive:
- For $$\ln(x-8)$$, we require $$x-8 > 0 \Rightarrow x > 8$$.
- For $$\ln(x+1)$$, we require $$x+1 > 0 \Rightarrow x > -1$$.
- For $$\ln(x-24)$$, we require $$x-24 > 0 \Rightarrow x > 24$$.

Thus, the overall domain is $$x > 24$$.

**Step 2. Rewrite the Equation without Logarithms**

Given:
$$
\ln(x-8) + \ln(x+1) = \ln(x-24)
$$
Using the logarithm property $$\ln(a) + \ln(b) = \ln(ab)$$, the equation becomes:
$$
\ln((x-8)(x+1)) = \ln(x-24)
$$
Removing the logarithms (assuming the arguments are in the domain) yields:
$$
(x-8)(x+1) = x-24
$$

**Step 3. Solve the Equation**

Expand the left-hand side:
$$
(x-8)(x+1) = x^2 + x - 8x - 8 = x^2 - 7x - 8
$$
Thus, the equation is:
$$
x^2 - 7x - 8 = x - 24
$$
Subtract $$x-24$$ from both sides:
$$
x^2 - 7x - 8 - x + 24 = 0 \quad \Longrightarrow \quad x^2 - 8x + 16 = 0
$$
Factor the quadratic:
$$
x^2 - 8x + 16 = (x-4)^2 = 0
$$
So, the potential solution is:
$$
x = 4
$$

**Step 4. Check the Domain**

The solution $$x = 4$$ must satisfy the domain requirement $$x > 24$$. Since $$4$$ is not greater than $$24$$, it is extraneous.

**Final Answer**

There is no solution that satisfies both the equation and its domain restrictions.

C. There is no solution.
There is no solution.

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