Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )$
A) $\{ - 3 \}$
B) $\{ 3 \}$
C) $\{ 3 , - 3 \}$
D) $\varnothing$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, we can combine the left side: $\ln((x-6)(x+1)) = \ln(x-15)$. This implies $(x-6)(x+1) = x-15$. Expanding and simplifying gives $x^2 - 5x - 6 = x - 15$, which simplifies to $x^2 - 6x + 9 = 0$, or $(x-3)^2 = 0$. The solution to this equation is $x = 3$. However, substituting $x = 3$ back into the original logarithmic expressions, we find that $\ln(3-6)$ and $\ln(3-15)$ are not defined because the logarithm of a negative number is not real. Therefore, there are no solutions that are in the domain of the original logarithmic expressions.

Solve the Logarithmic Equation ln⁡(x−6)+ln⁡(x+1)=ln⁡(x−15)\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )ln(x−6)+ln(x+1)=ln(x−15)

Solve the Logarithmic Equation $\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )$