Answer :
To determine whether the function [tex]\( F(x) = \log_5{x} \)[/tex] is decreasing, we need to understand the properties of logarithmic functions, particularly those with a base greater than 1.
1. Definition of a Logarithmic Function with Base [tex]\( 5 \)[/tex]:
The function [tex]\( F(x) = \log_5{x} \)[/tex] represents the logarithm of [tex]\( x \)[/tex] with base 5. This can be rewritten using the fact that logarithms and exponentials are inverses: [tex]\( \log_b{x} = y \)[/tex] if and only if [tex]\( b^y = x \)[/tex].
2. Behavior of Logarithms with Base Greater than 1:
A fundamental property of logarithms is that if the base [tex]\( b \)[/tex] is greater than 1, then the logarithmic function [tex]\( \log_b{x} \)[/tex] is an increasing function. That means as [tex]\( x \)[/tex] increases, [tex]\( \log_b{x} \)[/tex] also increases.
3. Visualization:
If we were to graph [tex]\( F(x) = \log_5{x} \)[/tex], we would see that it rises from [tex]\( -\infty \)[/tex] to [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] increases from 0 to [tex]\( +\infty \)[/tex]. This confirms that [tex]\( F(x) \)[/tex] is an increasing function.
4. Conclusion:
Since [tex]\( F(x) = \log_5{x} \)[/tex] is increasing for bases greater than 1, the statement that the function [tex]\( F(x) = \log_5{x} \)[/tex] is decreasing is false.
Thus, the correct answer is:
B. False
1. Definition of a Logarithmic Function with Base [tex]\( 5 \)[/tex]:
The function [tex]\( F(x) = \log_5{x} \)[/tex] represents the logarithm of [tex]\( x \)[/tex] with base 5. This can be rewritten using the fact that logarithms and exponentials are inverses: [tex]\( \log_b{x} = y \)[/tex] if and only if [tex]\( b^y = x \)[/tex].
2. Behavior of Logarithms with Base Greater than 1:
A fundamental property of logarithms is that if the base [tex]\( b \)[/tex] is greater than 1, then the logarithmic function [tex]\( \log_b{x} \)[/tex] is an increasing function. That means as [tex]\( x \)[/tex] increases, [tex]\( \log_b{x} \)[/tex] also increases.
3. Visualization:
If we were to graph [tex]\( F(x) = \log_5{x} \)[/tex], we would see that it rises from [tex]\( -\infty \)[/tex] to [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] increases from 0 to [tex]\( +\infty \)[/tex]. This confirms that [tex]\( F(x) \)[/tex] is an increasing function.
4. Conclusion:
Since [tex]\( F(x) = \log_5{x} \)[/tex] is increasing for bases greater than 1, the statement that the function [tex]\( F(x) = \log_5{x} \)[/tex] is decreasing is false.
Thus, the correct answer is:
B. False
