To find the vertical asymptote of the logarithmic function [tex]\( f(x) = \log_{\frac{1}{3}}(x) \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches certain values.
For any logarithmic function of the form [tex]\( f(x) = \log_b(x) \)[/tex], we must remember that the logarithm is undefined for non-positive values of [tex]\( x \)[/tex]. This means that [tex]\( x \)[/tex] must be greater than 0. Because of this, the logarithmic function will have a vertical asymptote at [tex]\( x = 0 \)[/tex].
The reason for this is that as [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( \log_b(x) \)[/tex] approaches negative infinity, which is a characteristic behavior of logarithmic functions. No matter what the base [tex]\( b \)[/tex] is (as long as it is between 0 and 1 or greater than 1), the function will not exist for [tex]\( x \leq 0 \)[/tex] and a vertical asymptote will always be present at [tex]\( x = 0 \)[/tex].
Therefore, for the given function [tex]\( f(x) = \log_{\frac{1}{3}}(x) \)[/tex], the vertical asymptote is at:
[tex]\[ x = 0 \][/tex]
