Find the domain of the logarithmic function.
-$$f ( x ) = \ln \left( \frac { 1 } { x - 6 } \right)$$
A) $( 6 , \infty )$
B) $( - 6 , \infty )$
C) $( 0 , \infty )$
D) $( 1 , \infty )$

Question

Quizplus · Accepted Answer

For the function $f(x) = \ln\left(\frac{1}{x - 6}\right)$ to be defined, the argument of the logarithm, $\frac{1}{x - 6}$, must be greater than zero. This means $x - 6$ must be less than zero (since the reciprocal of a positive number is positive, and the reciprocal of a negative number is negative). Therefore, $x < 6$, which corresponds to the interval $(-\infty, 6)$. However, the correct choice should indicate the domain where the argument of the logarithm is positive, not negative. The mistake in my explanation points to a reconsideration of the argument's condition for the logarithm to be defined: the argument $\frac{1}{x - 6}$ must be positive, which occurs when $x - 6$ is positive, not negative. This corrects the domain to $x > 6$, matching choice A: $(6, \infty)$.

Find the Domain of the Logarithmic Function f(x)=ln⁡(1x−6)f ( x ) = \ln \left( \frac { 1 } { x - 6 } \right)f(x)=ln(x−61​)

Find the Domain of the Logarithmic Function $f ( x ) = \ln \left( \frac { 1 } { x - 6 } \right)$