Question

What asymptote does a logarithmic function have? Why?

Answer 1

A logarithmic function, such as \( f(x) = \log_b(x) \) (where \( b > 0 \) and \( b \neq 1 \)), has a vertical asymptote at \( x = 0 \).

### Explanation:

1. **Domain of the Function**: The logarithmic function is defined only for positive values of \( x \). This means that as \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)), the function approaches negative infinity:
\[
\lim_{x \to 0^+} \log_b(x) = -\infty.
\]
The logarithmic function does not have any defined values for \( x \leq 0 \).

2. **Vertical Asymptote**: The presence of a vertical asymptote is indicated when the function approaches infinity (or negative infinity) as it gets closer to a particular value within its domain. Because \( f(x) = \log_b(x) \) approaches negative infinity as \( x \) approaches 0, we identify \( x = 0 \) as a vertical asymptote of the function.

### Summary:
Thus, the logarithmic function has a vertical asymptote at \( x = 0 \) because it is undefined for non-positive values and it trends towards negative infinity as \( x \) approaches 0 from the right.