Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.