Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithms
\ln (x - 6) + \ln (x + 3) = \ln (x - 22)

solve the equation to find the solution set. select the correct choice below and, if necessary, fill in the answer box

\bigcirc a. the solution set is \\{\\ \\}.
(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

Using the property \(\ln a+\ln b = \ln(ab)\), we rewrite the left - hand side of the equation \(\ln(x - 6)+\ln(x + 3)=\ln(x - 22)\) as \(\ln[(x - 6)(x + 3)]=\ln(x - 22)\).

Step2: Set Arguments Equal

Since the natural logarithm function \(y = \ln u\) is one - to - one (if \(\ln u=\ln v\), then \(u = v\) for \(u>0\) and \(v>0\)), we can set the arguments equal:
\((x - 6)(x + 3)=x - 22\)
Expand the left - hand side: \(x^{2}+3x-6x - 18=x - 22\)
Simplify to get a quadratic equation: \(x^{2}-3x - 18=x - 22\)
Move all terms to one side: \(x^{2}-3x - x-18 + 22 = 0\)
\(x^{2}-4x + 4 = 0\)

Step3: Solve the Quadratic Equation

The quadratic equation \(x^{2}-4x + 4 = 0\) is a perfect square trinomial, which can be factored as \((x - 2)^{2}=0\).
Taking the square root of both sides, we get \(x-2 = 0\), so \(x = 2\).

Step4: Check the Domain

Now we need to check the domain of the original logarithmic functions.
For \(\ln(x - 6)\) to be defined, \(x-6>0\Rightarrow x>6\).
For \(\ln(x + 3)\) to be defined, \(x + 3>0\Rightarrow x>- 3\).
For \(\ln(x - 22)\) to be defined, \(x-22>0\Rightarrow x>22\).
The value \(x = 2\) does not satisfy \(x>22\) (or \(x>6\)). So we reject \(x = 2\).

Answer:

C. There is no solution.