Answer :
To determine for what values of [tex]\( b \)[/tex] the function [tex]\( F(x) = \log_b(x) \)[/tex] is a decreasing function, we need to understand the behavior of the logarithm function with different bases.
### Properties of Logarithmic Functions
1. [tex]\( b > 1 \)[/tex]: When the base [tex]\( b \)[/tex] is greater than 1, the function [tex]\( \log_b(x) \)[/tex] is increasing. This means that as [tex]\( x \)[/tex] increases, [tex]\( \log_b(x) \)[/tex] also increases.
2. [tex]\( 0 < b < 1 \)[/tex]: When the base [tex]\( b \)[/tex] is a positive fraction less than 1, the function [tex]\( \log_b(x) \)[/tex] is decreasing. In this case, as [tex]\( x \)[/tex] increases, [tex]\( \log_b(x) \)[/tex] decreases.
To confirm which of the answer choices matches this condition:
- Choice A: [tex]\( 0 > b > -1 \)[/tex]
This specifies an interval of bases that are negative or zero, which is not relevant here since the base of a logarithmic function must be positive.
- Choice B: [tex]\( b < 0 \)[/tex]
This means the base is negative, which is not valid for the logarithmic function; the base of a log must be positive.
- Choice C: [tex]\( 0 < b < 1 \)[/tex]
This specifies that the base [tex]\( b \)[/tex] is between 0 and 1, which aligns exactly with our required condition for [tex]\( \log_b(x) \)[/tex] to be a decreasing function.
- Choice D: [tex]\( b > 0 \)[/tex]
This choice does not narrow it down sufficiently. While it includes positive bases, it includes bases greater than 1, which lead to the function being increasing rather than decreasing.
Therefore, the correct answer is:
[tex]\[ \boxed{0 < b < 1} \][/tex]
This means that [tex]\( F(x) = \log_b(x) \)[/tex] will be a decreasing function when the base [tex]\( b \)[/tex] is between 0 and 1.
### Properties of Logarithmic Functions
1. [tex]\( b > 1 \)[/tex]: When the base [tex]\( b \)[/tex] is greater than 1, the function [tex]\( \log_b(x) \)[/tex] is increasing. This means that as [tex]\( x \)[/tex] increases, [tex]\( \log_b(x) \)[/tex] also increases.
2. [tex]\( 0 < b < 1 \)[/tex]: When the base [tex]\( b \)[/tex] is a positive fraction less than 1, the function [tex]\( \log_b(x) \)[/tex] is decreasing. In this case, as [tex]\( x \)[/tex] increases, [tex]\( \log_b(x) \)[/tex] decreases.
To confirm which of the answer choices matches this condition:
- Choice A: [tex]\( 0 > b > -1 \)[/tex]
This specifies an interval of bases that are negative or zero, which is not relevant here since the base of a logarithmic function must be positive.
- Choice B: [tex]\( b < 0 \)[/tex]
This means the base is negative, which is not valid for the logarithmic function; the base of a log must be positive.
- Choice C: [tex]\( 0 < b < 1 \)[/tex]
This specifies that the base [tex]\( b \)[/tex] is between 0 and 1, which aligns exactly with our required condition for [tex]\( \log_b(x) \)[/tex] to be a decreasing function.
- Choice D: [tex]\( b > 0 \)[/tex]
This choice does not narrow it down sufficiently. While it includes positive bases, it includes bases greater than 1, which lead to the function being increasing rather than decreasing.
Therefore, the correct answer is:
[tex]\[ \boxed{0 < b < 1} \][/tex]
This means that [tex]\( F(x) = \log_b(x) \)[/tex] will be a decreasing function when the base [tex]\( b \)[/tex] is between 0 and 1.
