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To solve the given problem and help graph the logarithmic function \( g(x) = \log_4 (x - 1) \), let's break it down step-by-step:

### 1. **Determine the Domain**:
The argument of the logarithmic function must be positive. So, for \( g(x) = \log_4(x - 1) \), we need:
\[
x - 1 > 0 \implies x > 1
\]
Thus, the domain of \( g(x) \) is \( (1, \infty) \).

### 2. **Determine the Range**:
Logarithmic functions have a range of all real numbers because they can take any value as the input increases:
\[
\text{Range} = (-\infty, \infty)
\]

### 3. **Asymptote**:
Since \( g(x) \) has a transformation of \( x - 1 \), the vertical asymptote is at \( x = 1 \).

### 4. **Plotting Points**:
We can find some key points to plot the graph:
- Choose \( x = 2 \):
  \[
  g(2) = \log_4(2 - 1) = \log_4(1) = 0
  \]
  So, \( (2, 0) \) is a point on the graph.
  
- Choose \( x = 5 \):
  \[
  g(5) = \log_4(5 - 1) = \log_4(4) = 1
  \]
  So, \( (5, 1) \) is another point on the graph.

### 5. **Graph**:
- Plot the points \( (2, 0) \) and \( (5, 1) \) on the graph.
- Draw the vertical asymptote at \( x = 1 \).
- The curve approaches the asymptote as \( x \to 1^+ \) and increases slowly as \( x \) increases.

### Final Answer:
- **Domain**: \( (1, \infty) \)
- **Range**: \( (-\infty, \infty) \)

Would you like further details on any specific step, or do you have additional questions?

Here are 5 related questions to explore:
1. How does the base of the logarithm affect the graph's shape?
2. What happens if you apply a vertical or horizontal shift to the function \( g(x) \)?
3. How does changing the argument inside the logarithm affect the asymptote?
4. How can you convert between logarithmic and exponential forms for solving?
5. How would the graph of \( \log_4(x + 1) \) differ from \( \log_4(x - 1) \)?

**Tip**: When graphing logarithmic functions, always check the vertical asymptote by solving the inequality inside the logarithm.

Graph the logarithmic function g(x) = log4(x - 1). To do this, plot two points on the graph of the function, and also draw the asymptote. Additionally, give the domain and range of the function using interval notation.

Learn how to graph the logarithmic function g(x) = log4(x - 1). This problem involves plotting points, drawing the vertical asymptote, and determining the domain and range using interval notation. Suitable for students in Grades 10-12.

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