Solve the equation.
-$$\log _ { 3 } ( x - 1 ) + \log _ { 3 } ( x - 7 ) = 3$$
A)-2
B)10
C)10, -2
D)11

Question

Quizplus · Accepted Answer

The equation can be rewritten using the property of logarithms: $\log_3((x-1)(x-7)) = 3$. This implies $(x-1)(x-7) = 3^3 = 27$. Solving the quadratic equation $x^2 - 8x + 7 = 27$ gives $x^2 - 8x - 20 = 0$. The solutions are $x = 10$ and $x = -2$. However, $\log_3(x-1)$ and $\log_3(x-7)$ are only defined for positive arguments, so $x-1 > 0$ and $x-7 > 0$, which means $x > 7$. Therefore, the only valid solution is $x = 10$.

Solve the Equation log⁡3(x−1)+log⁡3(x−7)=3\log _ { 3 } ( x - 1 ) + \log _ { 3 } ( x - 7 ) = 3log3​(x−1)+log3​(x−7)=3

Solve the Equation $\log _ { 3 } ( x - 1 ) + \log _ { 3 } ( x - 7 ) = 3$