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: $\ln(x + 1) - \ln(x - 9) = \ln\left(\frac{x + 1}{x - 9}\right)$. This gives us $\ln(x - 5) = \ln\left(\frac{x + 1}{x - 9}\right)$. Since the logarithmic functions are equal, their arguments must be equal: $x - 5 = \frac{x + 1}{x - 9}$. Solving this equation for $x$ yields $x = 4$ and $x = 11$, which are the solutions that satisfy the original equation.

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 $\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )$