To match the logarithmic function $f(x) = \ln(x - 1)$ with the given graphs, we need to carefully analyze its properties and compare them to the description of graph $I$.

Step 1: Key properties of $f(x) = \ln(x - 1)$

1. Domain: The logarithmic function $\ln(x - 1)$ is defined only for $x - 1 > 0$, meaning $x > 1$. The function has a vertical asymptote at $x = 1$.
2. Behavior near $x = 1$: As $x \to 1^+$, $\ln(x - 1) \to -\infty$. The graph approaches the line $x = 1$ but never touches it.
3. Intercept: The $x$-intercept occurs when $\ln(x - 1) = 0$, which happens when $x - 1 = 1$, or $x = 2$.
4. Increasing nature: The function is increasing for all $x > 1$, as the natural logarithm is an increasing function.
5. Point to check: At $x = 3$, for example, $\ln(3 - 1) = \ln(2) \approx 0.693$.

Step 2: Analyze graph $I$

1. Graph $I$ enters the window almost vertically to the right of the $y$-axis.
2. It becomes less steep as $x$ increases and exits the window in the first quadrant.
3. The graph crosses the $x$-axis at approximately $x = 0.37$.
4. The graph passes through the point $(1, 1)$.

Step 3: Match properties with graph $I$

• The function $f(x) = \ln(x - 1)$ is undefined for $x \leq 1$, so it cannot pass through $(1, 1)$, making graph $I$ inconsistent with the given function.

Step 4: Conclusion

The function $f(x) = \ln(x - 1)$ does not match graph I because graph $I$ suggests a logarithmic function of the form $\ln(x + c)$ where $c > 0$, or another shifted logarithmic function.

Would you like a detailed explanation of why $f(x)$ aligns with another type of graph, or have any specific follow-ups?

Questions:

1. What transformations occur if the logarithmic function is shifted horizontally?
2. How does the domain of logarithmic functions affect their graph?
3. Can you describe the significance of the vertical asymptote in logarithmic functions?
4. How does the behavior near the asymptote differ for $\ln(x - c)$ versus $\ln(c - x)$?
5. What is the role of the $x$-intercept in identifying the correct logarithmic graph?

Tip:

Always check the domain and intercepts of a logarithmic function before attempting to match it to a graph!