To determine whether the function [tex]\( F(x) = \log_5(x) \)[/tex] is decreasing, we need to understand the properties of logarithmic functions and their behavior.
Let's start with some basic properties of logarithmic functions:
1. A logarithmic function [tex]\( \log_b(x) \)[/tex] has a base [tex]\( b \)[/tex], which is a positive real number.
2. The base [tex]\( b \)[/tex] of a logarithmic function determines the function's behavior:
- If [tex]\( b > 1 \)[/tex], the logarithmic function is increasing.
- If [tex]\( 0 < b < 1 \)[/tex], the logarithmic function is decreasing.
For the function [tex]\( F(x) = \log_5(x) \)[/tex], the base is 5. Since 5 is greater than 1, the following property holds:
- The function [tex]\( \log_5(x) \)[/tex] is increasing, not decreasing.
Therefore, the statement "The function [tex]\( F(x) = \log_5(x) \)[/tex] is decreasing" is evaluated as follows:
Given that the base 5 is greater than 1, [tex]\( F(x) = \log_5(x) \)[/tex] is an increasing function.
Thus, the correct answer is:
B. False
