Question

graphing logarithmic functions quick check
where does the logarithmic function $y = \log_{4} x$ have a vertical asymptote? (1 point)
\\(\circ\\) $x = 0$
\\(\circ\\) $y = 4$
\\(\circ\\) $x = 4$
\\(\circ\\) the logarithmic function $y = \log_{4} x$ has no vertical asymptote.

Explanation:

For a logarithmic function of the form \( y = \log_b x \) (where \( b>0, b
eq1 \)), the domain is \( x>0 \). As \( x \) approaches 0 from the right (\( x
ightarrow0^+ \)), the function values tend to \( -\infty \) (or \( +\infty \) depending on the base, but the key is the behavior near \( x = 0 \)). A vertical asymptote occurs where the function approaches infinity (or negative infinity) and the function is undefined there. For \( y=\log_4 x \), as \( x \) approaches 0, the function behaves such that \( x = 0 \) is a vertical asymptote. The option \( y = 4 \) is a horizontal line, not a vertical asymptote. \( x = 4 \) is just a point on the x - axis, not an asymptote. And we know logarithmic functions of the form \( \log_b x \) do have a vertical asymptote at \( x = 0 \).

Answer:

A. \( x = 0 \)