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.