If f(x)
 is the logarithmic parent function, which statement best describes how its intercepts change when transformed to g(x)=a⋅f(x)
 for some a∈R?
A.The x-intercepts stay the same but the y-intercepts are multiplied by a.
B.The y-intercepts stay the same but the x-intercepts are multiplied by a.
C.The y-intercepts are increased by a; the x-intercepts change depending on the shape of the function.
D.The x-intercepts are increased by a; the y-intercepts change depending on the shape of the function.

Given that \( f(x) \) is the logarithmic parent function, the question is asking how the intercepts change when it is transformed to \( g(x) = a \cdot f(x) \), where \( a \in \mathbb{R} \).

### Key Points:
1. **Logarithmic Parent Function:** Typically, \( f(x) = \log_b(x) \), where \( b > 0 \) and \( b \neq 1 \).
2. **Intercepts:**
   - **x-intercept:** The x-intercept of \( f(x) = \log_b(x) \) is \( (1, 0) \) because \( \log_b(1) = 0 \).
   - **y-intercept:** There is no y-intercept for the logarithmic parent function because \( \log_b(x) \) is undefined for \( x \leq 0 \).

### Transformation \( g(x) = a \cdot f(x) \):
- **x-intercept:** The x-intercept remains unchanged at \( x = 1 \), regardless of the value of \( a \), because \( g(1) = a \cdot \log_b(1) = a \cdot 0 = 0 \).
- **y-intercept:** Since there is no y-intercept for \( f(x) \), multiplying by \( a \) does not introduce a y-intercept either.

### Conclusion:
The x-intercepts stay the same, but the y-values (which correspond to the y-intercept in functions that have them) are multiplied by \( a \). This matches the scenario described in option **A**.

### Correct Answer:
**A. The x-intercepts stay the same but the y-intercepts are multiplied by a.**

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### Related Questions:
1. How does the graph of \( f(x) = \log_b(x) \) change when \( b \) is altered?
2. What happens to the intercepts if the logarithmic function is shifted horizontally?
3. Can a logarithmic function have a y-intercept? Why or why not?
4. How does a vertical stretch affect the graph of \( f(x) = \log_b(x) \)?
5. What are the asymptotes of the logarithmic parent function and how do they transform under \( g(x) = a \cdot f(x) \)?

### Tip:
When analyzing transformations of functions, always consider how each transformation (stretch, shift, etc.) affects key points like intercepts and asymptotes.

If f(x) is the logarithmic parent function, which statement best describes how its intercepts change when transformed to g(x)=a⋅f(x) for some a∈R?

Explore how the intercepts of the logarithmic parent function f(x) = log_b(x) change when transformed to g(x) = a⋅f(x). This problem is suitable for high school students in Grades 10-12 and covers key concepts such as logarithmic functions, their intercepts, and the effects of transformations.

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