Solve the logarithmic equation below algebraically.
$$\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )$$
A) $- 12$ and $- 9$
B) 4 and 11
C) $- 12$ and 11
D) $- 9$ and 11
E) $- 12$ and 4

Question

Quizplus · Accepted Answer

Using the properties of logarithms, we can combine the right side of the equation using the quotient rule: $\ln(x+1) - \ln(x-9) = \ln\left(\frac{x+1}{x-9}\right)$. This gives us $\ln(x-5) = \ln\left(\frac{x+1}{x-9}\right)$. Since the natural log function is one-to-one, we can equate the arguments: $x-5 = \frac{x+1}{x-9}$. Solving this equation for $x$ involves clearing the denominator and finding the roots of the resulting quadratic equation, which yields $x = 4$ and $x = 11$.

Solve the Logarithmic Equation Below Algebraically ln⁡(x−5)=ln⁡(x+1)−ln⁡(x−9)\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )ln(x−5)=ln(x+1)−ln(x−9)

Solve the Logarithmic Equation Below Algebraically $\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )$