Question

evaluate y = ln(x - 2) for the following values of x. round to the nearest thousandth. this is a multi - part item. which of the following is the graph of y = ln(x - 2)? a: x = 3, y = 0 <
b: x = 4, y = 0.693 <
c: x = 6, y = 1.386 <

Explanation:

Step1: Recall domain of natural - log function

The domain of $y = \ln(x - 2)$ is $x-2>0$, i.e., $x > 2$. The graph of $y=\ln(x)$ is shifted 2 units to the right.

Step2: Analyze vertical asymptote

The vertical asymptote of $y = \ln(x - 2)$ is $x = 2$. As $x$ approaches 2 from the right, $y\to-\infty$, and as $x\to+\infty$, $y\to+\infty$.

Step3: Evaluate for given $x$ - values

For $x = 3$, $y=\ln(3 - 2)=\ln(1)=0$.
For $x = 4$, $y=\ln(4 - 2)=\ln(2)\approx0.693$.
For $x = 6$, $y=\ln(6 - 2)=\ln(4)\approx1.386$.

Answer:

For the graph question, the correct graph is the one with a vertical asymptote at $x = 2$ and passing through points that follow the behavior of the natural - log function shifted 2 units to the right.
For the evaluations:
When $x = 3$, $y = 0$.
When $x = 4$, $y\approx0.693$.
When $x = 6$, $y\approx1.386$.