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 - 3) - \ln (x + 4) = \ln (\ln (x - 2) - \ln (x + 9)\\) 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 \\(\square\\). (type an integer or a simplified fraction. use a comma to separate answers as needed.) \\(\bigcirc\\) b. there are infinitely many solutions. \\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Apply Logarithmic Quotient Rule

The logarithmic equation is \(\ln(x - 3)-\ln(x + 4)=\ln(x - 2)-\ln(x + 9)\). Using the quotient rule of logarithms \(\ln a-\ln b = \ln\frac{a}{b}\), we can rewrite both sides:
\(\ln\frac{x - 3}{x + 4}=\ln\frac{x - 2}{x + 9}\)

Step2: Use One - to - One Property of Logarithms

Since the natural logarithm function \(y = \ln x\) is one - to - one (if \(\ln a=\ln b\), then \(a = b\) for \(a>0,b>0\)), we can set the arguments equal to each other:
\(\frac{x - 3}{x + 4}=\frac{x - 2}{x + 9}\)

Step3: Cross - Multiply to Eliminate Fractions

Cross - multiplying gives \((x - 3)(x + 9)=(x - 2)(x + 4)\)
Expand both sides:
\(x^{2}+9x-3x - 27=x^{2}+4x-2x - 8\)
Simplify both sides:
\(x^{2}+6x - 27=x^{2}+2x - 8\)

Step4: Solve for \(x\)

Subtract \(x^{2}\) from both sides:
\(6x-27 = 2x-8\)
Subtract \(2x\) from both sides:
\(4x-27=-8\)
Add 27 to both sides:
\(4x=19\)
Divide both sides by 4:
\(x=\frac{19}{4}=4.75\)

Step5: Check the Domain

For the original logarithmic expressions:

• For \(\ln(x - 3)\), we need \(x-3>0\Rightarrow x > 3\)
• For \(\ln(x + 4)\), we need \(x + 4>0\Rightarrow x>-4\)
• For \(\ln(x - 2)\), we need \(x-2>0\Rightarrow x > 2\)
• For \(\ln(x + 9)\), we need \(x + 9>0\Rightarrow x>-9\)

The value \(x = \frac{19}{4}=4.75\) satisfies \(x>3\), \(x>-4\), \(x > 2\) and \(x>-9\).

Answer:

\(\frac{19}{4}\)