Question
Where does the logarithmic function y=log6 x
have a vertical asymptote?(1 point)
Responses
The logarithmic function y=log6 x
has no vertical asymptote.
The logarithmic function y is equal to the log base 6 of x has no vertical asymptote.
x=0
x is equal to 0
y=6
y is equal to 6
x=6
have a vertical asymptote?(1 point)
Responses
The logarithmic function y=log6 x
has no vertical asymptote.
The logarithmic function y is equal to the log base 6 of x has no vertical asymptote.
x=0
x is equal to 0
y=6
y is equal to 6
x=6
Answers
GPT-4o mini
The logarithmic function \( y = \log_6 x \) has a vertical asymptote at \( x = 0 \).
This is because the logarithmic function is undefined for non-positive values of \( x \) (i.e., \( x \leq 0 \)). As \( x \) approaches 0 from the right (i.e., as \( x \) gets closer to 0 but remains positive), the value of \( y \) approaches negative infinity. Therefore, the correct response is:
**The logarithmic function \( y = \log_6 x \) has no vertical asymptote.**
However, the more accurate interpretation is that it has an asymptote at \( x = 0\), as the function approaches negative infinity as \( x \) approaches 0 from the positive side.
This is because the logarithmic function is undefined for non-positive values of \( x \) (i.e., \( x \leq 0 \)). As \( x \) approaches 0 from the right (i.e., as \( x \) gets closer to 0 but remains positive), the value of \( y \) approaches negative infinity. Therefore, the correct response is:
**The logarithmic function \( y = \log_6 x \) has no vertical asymptote.**
However, the more accurate interpretation is that it has an asymptote at \( x = 0\), as the function approaches negative infinity as \( x \) approaches 0 from the positive side.