Question

Describe the horizontal and vertical asymptotes of f(x)=log4x
A. The function f(x) has a horizontal asymptote at y=0 and no vertical asymptote.
B. The function f(x) has a vertical asymptote at x=0 and no horizontal asymptote.
C. The function f(x) has a horizontal asymptote at y=4 and a vertical asymptote at x=0 .
D. The function f(x) has a horizontal asymptote at y=0 and a vertical asymptote at x=4 .

Answer 1

To analyze the function \( f(x) = \log_4(x) \) for its asymptotic behavior, we need to consider its properties regarding horizontal and vertical asymptotes:

1. **Vertical Asymptote**: The function \( \log_4(x) \) is defined only for \( x > 0 \). As \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)), \( f(x) \) approaches \(-\infty\). Therefore, there is a vertical asymptote at \( x = 0 \).

2. **Horizontal Asymptote**: As \( x \) goes to infinity (i.e., \( x \to \infty \)), \( \log_4(x) \) goes to infinity as well. Thus, there is no horizontal asymptote, as the function does not approach a finite value.

Based on this analysis, the correct conclusion is:

**B. The function f(x) has a vertical asymptote at x=0 and no horizontal asymptote.**