To determine the values of [tex]\( b \)[/tex] for which the function [tex]\( F(x) = \log_b(x) \)[/tex] is a decreasing function, we need to understand the behavior of logarithmic functions.
1. A logarithmic function [tex]\( F(x) = \log_b(x) \)[/tex] is defined only for positive values of [tex]\( b \)[/tex] other than 1 ([tex]\( b > 0 \)[/tex] and [tex]\( b \neq 1 \)[/tex]).
2. The behavior of the function depends on the base [tex]\( b \)[/tex]:
- If [tex]\( b > 1 \)[/tex], the function [tex]\( F(x) = \log_b(x) \)[/tex] is an increasing function. As [tex]\( x \)[/tex] increases, [tex]\( \log_b(x) \)[/tex] also increases.
- If [tex]\( 0 < b < 1 \)[/tex], the function [tex]\( F(x) = \log_b(x) \)[/tex] is a decreasing function. As [tex]\( x \)[/tex] increases, [tex]\( \log_b(x) \)[/tex] decreases.
To summarize, for [tex]\( F(x) = \log_b(x) \)[/tex] to be a decreasing function, the base [tex]\( b \)[/tex] must be between 0 and 1. Therefore, the correct condition is:
[tex]\[ 0 < b < 1 \][/tex]
This corresponds to option B.
Thus, the correct answer is:
B. [tex]\( 0 < b < 1 \)[/tex]
