Question

evaluate $y = \ln (x - 2)$ for the following values of $x$. round to the nearest thousandth.\
$x = 3, y = \boxed{0}$\
$x = 4, y = \boxed{0.693}$\
$x = 6, y = \boxed{1.386}$\
which of the following is the graph of $y = \ln (x - 2)$?\
three graphs are shown here, each with a coordinate plane and a curve. the first graph has y-axis labeled 6,4,2,-2 and x-axis with a curve; the second has y-axis labeled 2,-2,-4 and x-axis with a curve; the third has y-axis labeled 4,2,-2 and x-axis with a curve, each with a radio button next to them.

Explanation:

Step1: Recall the domain of \( y = \ln(x - 2) \)

The natural logarithm function \( \ln(u) \) is defined when \( u>0 \). So for \( y=\ln(x - 2) \), we need \( x - 2>0\), which means \( x>2 \). So the domain is \( x\in(2,+\infty) \). This tells us the graph should only exist for \( x>2 \).

Step2: Analyze the vertical asymptote

The vertical asymptote of \( y = \ln(x - 2) \) occurs where \( x - 2 = 0\), i.e., \( x = 2 \). As \( x\) approaches \( 2 \) from the right (\( x
ightarrow2^+ \)), \( y=\ln(x - 2)
ightarrow-\infty \).

Step3: Analyze the behavior as \( x

ightarrow+\infty \)
As \( x
ightarrow+\infty \), \( x - 2
ightarrow+\infty \), and \( \ln(x - 2)
ightarrow+\infty \), so the function is increasing.

Step4: Check the graphs

• First graph: Let's check the \( x\)-values. The graph seems to be for \( x>2 \)? Wait, looking at the axes, the first graph's \( x\)-axis: let's see the grid. Wait, the second graph: wait, no, let's re - examine. Wait, the first graph (top one): the curve is on the left of \( x = 2 \)? No, wait the axes: maybe I misread. Wait, the function \( y=\ln(x - 2) \) is a horizontal shift of \( y = \ln(x) \) to the right by 2 units. The graph of \( y=\ln(x) \) has domain \( x>0 \), passes through \( (1,0) \), vertical asymptote \( x = 0 \), increasing. So \( y=\ln(x - 2) \) has domain \( x>2 \), passes through \( (3,0) \) (since when \( x = 3 \), \( y=\ln(1)=0 \)), vertical asymptote \( x = 2 \), increasing.

Now let's check the three graphs:

• Top graph: Let's see the \( x\)-intercept. If \( y = 0 \), then \( \ln(x - 2)=0\Rightarrow x - 2 = 1\Rightarrow x = 3 \). So the graph should pass through \( (3,0) \). Let's check the top graph: does it pass through \( (3,0) \)? Looking at the grid, the top graph's curve: when \( x = 3 \), what's \( y \)? Wait, maybe the axes: the top graph's \( y\)-axis has 6,4,2, - 2? Wait, no, maybe the first graph (top) has \( x\) - axis with positive direction to the right, and the curve is for \( x>2 \)? Wait, no, the second graph: let's check the third graph (bottom). The bottom graph has a curve that seems to be for \( x<0 \) or something, no. Wait, the middle graph: no, wait the top graph: let's check the key points. The function \( y=\ln(x - 2) \) passes through \( (3,0) \), \( (e + 2,1)\approx(4.718,1) \), \( (e^2+2,2)\approx(9.389,2) \).

Wait, the top graph: when \( x = 3 \), \( y = 0 \) (since \( \ln(3 - 2)=\ln(1)=0 \)). Let's check the top graph's coordinates. If the grid is such that the intersection of the axes is at some point, and the top graph has a curve that passes through \( (3,0) \) (since when \( x = 3 \), \( y = 0 \)), and the domain \( x>2 \). Let's check the other graphs:

• Middle graph: The curve seems to be for \( x<2 \) (since the \( x\)-values are less than 2), which is outside the domain of \( y=\ln(x - 2) \) (domain \( x>2 \)), so middle graph is out.

• Bottom graph: The curve is in the region where \( x\) is positive but maybe \( x<2 \), and the direction is different.

The top graph: Let's confirm the vertical asymptote. As \( x
ightarrow2^+ \), \( y
ightarrow-\infty \), which matches the top graph's behavior (the curve goes down towards \( x = 2 \) from the right). And as \( x \) increases, \( y \) increases (since natural log is increasing), which also matches the top graph (the curve goes up as \( x \) increases to the right). Also, when \( x = 3 \), \( y = 0 \), which should be on the graph, and the top graph seems to pass through \( (3,0) \) (from the grid).

Answer:

The Top Graph (the first graph from the top)