To determine the range of the function \( y = \log_3(x - 4) \), let’s analyze its properties:

### Step-by-Step Solution

1. **Identify the Domain of \( y = \log_3(x - 4) \):**
   - For the function \( y = \log_3(x - 4) \) to be defined, the argument inside the logarithm must be positive:
     \[
     x - 4 > 0
     \]
     \[
     x > 4
     \]
   - So, the domain of the function is \( x > 4 \).

2. **Analyze the Behavior of \( y = \log_3(x - 4) \):**
   - As \( x \to 4^+ \) (approaches 4 from the right), \( x - 4 \to 0^+ \), so \( \log_3(x - 4) \to -\infty \).
   - As \( x \to \infty \), \( x - 4 \to \infty \), so \( \log_3(x - 4) \to \infty \).
   
3. **Determine the Range of \( y = \log_3(x - 4) \):**
   - Since \( y \) can take any real value from \( -\infty \) to \( \infty \) as \( x \) varies over its domain, the range of \( y = \log_3(x - 4) \) is:
     \[
     \text{Range of } y = (-\infty, \infty)
     \]

### Final Answer
The range of \( y = \log_3(x - 4) \) is:
\[
(-\infty, \infty)
\]

Would you like further details on this solution or have additional questions?

#### Related Questions
1. What is the domain of \( y = \log_3(x - 4) \)?
2. How would the range change if we used \( y = \log_3(x + 4) \) instead?
3. What happens to the range if we have \( y = -\log_3(x - 4) \)?
4. How would the base of the logarithm (e.g., changing to \( \log_2 \)) affect the range?
5. Can you graph \( y = \log_3(x - 4) \) to visualize its range?

#### Tip:
For any logarithmic function \( y = \log_b(x - c) \), the range is always \( (-\infty, \infty) \), while the domain depends on making \( x - c \) positive.

Discover how to find the range of the function y = log3(x - 4) with a detailed, step-by-step solution. This problem explores logarithmic functions, domain and range, and properties of logarithmic behavior. Ideal for high school students in Grades 10-12.

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