To determine whether the function [tex]\( F(x) = \log_{0.75} x \)[/tex] is decreasing or not, we need to consider the properties of logarithmic functions and their bases.
1. Understanding Logarithmic Functions:
A logarithmic function [tex]\( \log_a(x) \)[/tex] has different behavior based on the base [tex]\(a\)[/tex]:
- If [tex]\( a > 1 \)[/tex], the function [tex]\( \log_a(x) \)[/tex] is increasing.
- If [tex]\( 0 < a < 1 \)[/tex], the function [tex]\( \log_a(x) \)[/tex] is decreasing.
2. Identify the Base of the Function:
In our given function [tex]\( F(x) = \log_{0.75} x \)[/tex], the base of the logarithm is [tex]\( 0.75 \)[/tex].
3. Analyze the Base:
The base [tex]\( 0.75 \)[/tex] is less than 1 but greater than 0, i.e., [tex]\( 0 < 0.75 < 1 \)[/tex].
4. Conclusion on Monotonicity:
Since the base [tex]\( 0.75 \)[/tex] is in the interval [tex]\( (0, 1) \)[/tex], according to the properties of logarithmic functions:
- [tex]\( F(x) = \log_{0.75} x \)[/tex] is a decreasing function.
Therefore, the statement "The function [tex]\( F(x)=\log_{0.75} x \)[/tex] is decreasing" is true.
Answer: A. True
