Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer.\\
\\(\ln(x - 4) + \ln(x + 1) = \ln(x - 8)\\)\\
\\
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.\\
\\(\bigcirc\\) a. the solution set is \\(\boxed{}\\) (simplify your answer. use a comma to separate answers as needed.)\\
\\(\bigcirc\\) b. there are infinitely many solutions.\\
\\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Apply Logarithm Product Rule

The equation is \(\ln(x - 4)+\ln(x + 1)=\ln(x - 8)+\ln(x + 1)\). Using the product rule of logarithms \(\ln a+\ln b=\ln(ab)\), we can rewrite both sides:
Left side: \(\ln[(x - 4)(x + 1)]\)
Right side: \(\ln[(x - 8)(x + 1)]\)
So the equation becomes \(\ln[(x - 4)(x + 1)]=\ln[(x - 8)(x + 1)]\).

Step2: Use Logarithm One - to - One Property

If \(\ln a=\ln b\), then \(a = b\) (for \(a>0,b>0\)). So we set \((x - 4)(x + 1)=(x - 8)(x + 1)\).
Expand both sides:
Left side: \(x^{2}+x-4x - 4=x^{2}-3x - 4\)
Right side: \(x^{2}+x-8x - 8=x^{2}-7x - 8\)
So the equation is \(x^{2}-3x - 4=x^{2}-7x - 8\).

Step3: Solve the Quadratic - like Equation

Subtract \(x^{2}\) from both sides:
\(-3x - 4=-7x - 8\)
Add \(7x\) to both sides:
\(4x - 4=-8\)
Add 4 to both sides:
\(4x=-4\)
Divide both sides by 4:
\(x=-1\)

Step4: Check the Domain

For the logarithm function \(\ln u\), \(u>0\).

• For \(\ln(x - 4)\), \(x-4>0\Rightarrow x > 4\)
• For \(\ln(x + 1)\), \(x + 1>0\Rightarrow x>-1\)
• For \(\ln(x - 8)\), \(x-8>0\Rightarrow x > 8\)

When \(x=-1\), \(\ln(x - 4)=\ln(-5)\) (undefined), \(\ln(x - 8)=\ln(-9)\) (undefined). So \(x = - 1\) is not in the domain.

Now, let's go back to the original equation and simplify it by subtracting \(\ln(x + 1)\) from both sides (note that \(x
eq - 1\) because of the domain of \(\ln(x + 1)\)). We get \(\ln(x - 4)=\ln(x - 8)\). Then \(x-4=x - 8\), which simplifies to \(-4=-8\), a contradiction. So there is no solution.

Answer:

C. There is no solution.