Solve the equation. Write the solution set with the exact solutions.
-$\ln x + \ln ( x - 4 ) = \ln ( 5 x - 14 )$
A) $\{ 2 \}$
B) $\{ 7 \}$
C) $\{ 2,7 \}$
D){ }

Question

Solve the equation. Write the solution set with the exact solutions.
-$\ln x + \ln ( x - 4 ) = \ln ( 5 x - 14 )$
A) $\{ 2 \}$
B) $\{ 7 \}$
C) $\{ 2,7 \}$
D){  }

Quizplus · Accepted Answer

Using the properties of logarithms, we can combine the logarithms on the left side of the equation: $\ln x + \ln (x - 4) = \ln (x(x - 4)) = \ln (x^2 - 4x)$. Thus, the equation becomes $\ln (x^2 - 4x) = \ln (5x - 14)$. Since the logarithmic function is one-to-one, the arguments must be equal if their logarithms are equal, leading to the equation $x^2 - 4x = 5x - 14$. Rearranging gives $x^2 - 9x + 14 = 0$, which factors to $(x - 2)(x - 7) = 0$. Therefore, the solutions are $x = 2$ and $x = 7$. However, substituting $x = 2$ back into the original equation, we find that $\ln(2) + \ln(-2)$ is not valid since the logarithm of a negative number is undefined. Thus, the only valid solution is $x = 7$.

Solve the Equation ln⁡x+ln⁡(x−4)=ln⁡(5x−14)\ln x + \ln ( x - 4 ) = \ln ( 5 x - 14 )lnx+ln(x−4)=ln(5x−14)

Solve the Equation $\ln x + \ln ( x - 4 ) = \ln ( 5 x - 14 )$