To determine whether the function [tex]\( F(x) = \log_{0.75} x \)[/tex] is decreasing, we need to consider the properties of logarithmic functions. Here's a detailed, step-by-step solution:
1. Understand the Base of the Logarithmic Function:
- The given function is [tex]\( F(x) = \log_{0.75} x \)[/tex].
- The base of the logarithm here is [tex]\( 0.75 \)[/tex].
2. Logarithmic Function Properties:
- For logarithmic functions of the form [tex]\( \log_b x \)[/tex], the behavior of the function depends on the base [tex]\( b \)[/tex].
- If [tex]\( b > 1 \)[/tex], the function [tex]\( \log_b x \)[/tex] is increasing.
- If [tex]\( 0 < b < 1 \)[/tex], the function [tex]\( \log_b x \)[/tex] is decreasing.
3. Determine the Range of the Base:
- Here, the base [tex]\( b = 0.75 \)[/tex].
- We need to check if [tex]\( 0.75 \)[/tex] lies in between 0 and 1.
4. Conclusion Based on the Base:
- Since [tex]\( 0 < 0.75 < 1 \)[/tex], we can conclude that the function [tex]\( \log_{0.75} x \)[/tex] is indeed a decreasing function.
Therefore, the correct answer is:
A. True
