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?

The logarithmic parent function \( f(x) \) generally refers to \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm. The graph of this function has an intercept at \( (1, 0) \) on the x-axis.

Now, consider the transformation \( g(x) = a \cdot f(x) \), where \( a \in \mathbb{R} \).

### How the transformation affects the intercepts:
- **Vertical Scaling:** The multiplication by \( a \) results in a vertical scaling of the graph.
- **x-Intercept:** Since the transformation is purely vertical (multiplying the output), the x-intercept does not change. The x-intercept remains at \( (1, 0) \).
- **y-Intercept:** The logarithmic function does not have a y-intercept because it is undefined at \( x = 0 \).

### Conclusion:
The x-intercept of the logarithmic function \( f(x) \) remains unchanged when transformed to \( g(x) = a \cdot f(x) \). The best description is that **the intercepts do not change**.

### Do you want more details or have any questions?

Here are five related questions to explore further:
1. What happens to the graph of \( f(x) = \log_b(x) \) if you add a constant \( c \) after scaling, i.e., \( g(x) = a \cdot f(x) + c \)?
2. How does the base \( b \) of the logarithmic function affect the shape and intercepts of \( f(x) \)?
3. What are the effects of horizontal transformations on the intercepts of \( f(x) = \log_b(x) \)?
4. How would the intercepts change if the transformation was \( g(x) = f(x) + d \) instead of \( g(x) = a \cdot f(x) \)?
5. Can you describe the intercepts for the natural logarithm \( f(x) = \ln(x) \) and its transformations?

**Tip:** When analyzing transformations, it’s helpful to focus on whether the transformation affects the input (x-axis) or the output (y-axis) of the function to determine changes in intercepts.

Explore how transforming the logarithmic parent function affects its intercepts. Learn that the x-intercept remains unchanged when the function is scaled vertically. Suitable for high school students studying logarithmic functions and function transformations.

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