Question

sketch the logarithmic function g(x) = \log_{\frac{1}{15}} x. identify three points that lie on the graph, and determine the domain and the equation of any vertical asympt\
b. determine the domain of g.\
the domain of g is (0,\infty) .\
(type your answer in interval notation.).\
c. determine the equation(s) of any vertical asymptote(s). select the correct choice and, if necessary, fill in the answer box to complete your choice.\
\bigcirc a. \square (use a comma to separate answers as needed.)\
\bigcirc b. there are no vertical asymptotes.

Explanation:

Step1: Recall Logarithmic Function Properties

For a logarithmic function \( g(x) = \log_b x \), the domain is \( (0, \infty) \) because the argument \( x \) must be positive. Also, the vertical asymptote of \( \log_b x \) is \( x = 0 \) (the y - axis) as \( x \) approaches 0 from the right, the function values tend to \( \pm\infty \) (depending on the base \( b \)).

Step2: Analyze Vertical Asymptote for \( g(x)=\log_{\frac{1}{15}}x \)

Since \( g(x)=\log_{\frac{1}{15}}x \) is a logarithmic function with base \( \frac{1}{15} \), it follows the general properties of logarithmic functions. The vertical asymptote occurs where the argument \( x = 0 \), because as \( x \to 0^+ \), \( \log_{\frac{1}{15}}x \) will tend to \( +\infty \) (since the base \( 0<\frac{1}{15}<1 \), the function is decreasing, and as \( x \) gets closer to 0 from the right, the function values increase without bound). So the equation of the vertical asymptote is \( x = 0 \). For the multiple - choice question about vertical asymptotes, option A should be selected with \( x = 0 \) filled in (or if we consider the options, we know that logarithmic functions of the form \( \log_b x \) have a vertical asymptote at \( x = 0 \), so we choose option A and the equation is \( x = 0 \)).

Answer:

For part c, the correct choice is A. \( x = 0 \)