The Points \[$(0,1), (1,14)\$\], And \[$(2,196)\$\] All Lie On The Line Of The Exponential Function \[$f(x) = 14^x\$\]. On Which Logarithmic Function Do The Points \[$(14,1)\$\] And \[$(196,2)\$\] Lie?A.
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Introduction
The exponential function {f(x) = 14^x$}$ is a fundamental concept in mathematics, and understanding its properties is crucial for solving various problems. In this article, we will explore the relationship between the exponential function and logarithmic functions. We will use the points {(14,1)$}$ and {(196,2)$}$ to determine which logarithmic function they lie on.
Understanding Exponential Functions
An exponential function is a function of the form {f(x) = a^x$}$, where {a$}$ is a positive real number. The exponential function {f(x) = 14^x$}$ is a specific case of this form, where {a = 14$}$. This function has a base of 14 and grows exponentially as the input {x$}$ increases.
Properties of Exponential Functions
Exponential functions have several important properties that are useful for solving problems. One of the key properties is that the exponential function is one-to-one, meaning that each output value corresponds to a unique input value. This property is essential for solving problems involving exponential functions.
Logarithmic Functions
A logarithmic function is the inverse of an exponential function. If {f(x) = a^x$}$ is an exponential function, then the logarithmic function {f^{-1}(x) = \log_a(x)$}$ is its inverse. The logarithmic function {\log_a(x)$}$ returns the exponent to which the base {a$}$ must be raised to produce the input value {x$}$.
Relationship Between Exponential and Logarithmic Functions
The relationship between exponential and logarithmic functions is fundamental to understanding many mathematical concepts. The exponential function {f(x) = a^x$}$ and its inverse, the logarithmic function {f^{-1}(x) = \log_a(x)$}$, are closely related. In fact, they are inverse functions, meaning that they "undo" each other.
Using the Points to Determine the Logarithmic Function
We are given the points {(14,1)$}$ and {(196,2)$}$ and asked to determine which logarithmic function they lie on. To do this, we need to find the logarithmic function that passes through these points.
Finding the Logarithmic Function
To find the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$, we can use the fact that the logarithmic function is the inverse of the exponential function. We can start by finding the exponential function that passes through the points {(1,14)$}$ and {(2,196)$}$.
Finding the Exponential Function
The exponential function {f(x) = 14^x$}$ passes through the points {(1,14)$}$ and {(2,196)$}$. We can use this information to find the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$.
Using the Inverse Relationship
Since the logarithmic function is the inverse of the exponential function, we can use the inverse relationship to find the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$. We can start by finding the inverse of the exponential function {f(x) = 14^x$}$.
Finding the Inverse
To find the inverse of the exponential function {f(x) = 14^x$}$, we can swap the x and y values and solve for y. This gives us the inverse function {f^{-1}(x) = \log_{14}(x)$}$.
Determining the Logarithmic Function
Now that we have found the inverse of the exponential function, we can determine the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$. We can use the fact that the logarithmic function is the inverse of the exponential function to find the correct logarithmic function.
Conclusion
In this article, we used the points {(14,1)$}$ and {(196,2)$}$ to determine which logarithmic function they lie on. We found that the logarithmic function {\log_{14}(x)$}$ passes through these points. This demonstrates the relationship between exponential and logarithmic functions and how they can be used to solve problems.
Final Answer
The points {(14,1)$}$ and {(196,2)$}$ lie on the logarithmic function {\log_{14}(x)$}$.
Additional Information
The logarithmic function {\log_{14}(x)$}$ is a specific case of the logarithmic function {\log_a(x)$}$, where {a = 14$}$. This function has a base of 14 and returns the exponent to which the base 14 must be raised to produce the input value x.
Key Takeaways
- The exponential function {f(x) = 14^x$}$ is a fundamental concept in mathematics.
- The logarithmic function {\log_a(x)$}$ is the inverse of the exponential function {f(x) = a^x$}$.
- The points {(14,1)$}$ and {(196,2)$}$ lie on the logarithmic function {\log_{14}(x)$}$.
- The logarithmic function {\log_{14}(x)$}$ is a specific case of the logarithmic function {\log_a(x)$}$, where {a = 14$}$.
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Q: What is the relationship between exponential and logarithmic functions?
A: The exponential function {f(x) = a^x$}$ and its inverse, the logarithmic function {f^{-1}(x) = \log_a(x)$}$, are closely related. In fact, they are inverse functions, meaning that they "undo" each other.
Q: How do you find the logarithmic function that passes through a given point?
A: To find the logarithmic function that passes through a given point, you can use the fact that the logarithmic function is the inverse of the exponential function. You can start by finding the exponential function that passes through the point, and then find the inverse of that function.
Q: What is the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$?
A: The logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$ is {\log_{14}(x)$}$.
Q: What is the base of the logarithmic function {\log_{14}(x)$}$?
A: The base of the logarithmic function {\log_{14}(x)$}$ is 14.
Q: What is the inverse of the exponential function {f(x) = 14^x$}$?
A: The inverse of the exponential function {f(x) = 14^x$}$ is the logarithmic function {\log_{14}(x)$}$.
Q: How do you determine the logarithmic function that passes through a given point?
A: To determine the logarithmic function that passes through a given point, you can use the fact that the logarithmic function is the inverse of the exponential function. You can start by finding the exponential function that passes through the point, and then find the inverse of that function.
Q: What is the relationship between the points {(14,1)$}$ and {(196,2)$}$ and the logarithmic function {\log_{14}(x)$}$?
A: The points {(14,1)$}$ and {(196,2)$}$ lie on the logarithmic function {\log_{14}(x)$}$.
Q: How do you use the points {(14,1)$}$ and {(196,2)$}$ to determine the logarithmic function?
A: You can use the points {(14,1)$}$ and {(196,2)$}$ to determine the logarithmic function by finding the exponential function that passes through these points, and then finding the inverse of that function.
Q: What is the significance of the points {(14,1)$}$ and {(196,2)$}$ in determining the logarithmic function?
A: The points {(14,1)$}$ and {(196,2)$}$ are significant in determining the logarithmic function because they lie on the logarithmic function {\log_{14}(x)$}$.
Q: How do you apply the concept of inverse functions to determine the logarithmic function?
A: You can apply the concept of inverse functions to determine the logarithmic function by finding the exponential function that passes through the given points, and then finding the inverse of that function.
Q: What is the final answer to the problem of determining the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$?
A: The final answer to the problem of determining the logarithmic function that passes through the points {(14,1)$}$ and {(196,2)$}$ is the logarithmic function {\log_{14}(x)$}$.
Q: What is the significance of the logarithmic function {\log_{14}(x)$}$ in the context of the problem?
A: The logarithmic function {\log_{14}(x)$}$ is significant in the context of the problem because it is the inverse of the exponential function {f(x) = 14^x$}$ and passes through the points {(14,1)$}$ and {(196,2)$}$.
Q: How do you use the logarithmic function {\log_{14}(x)$}$ to solve problems involving exponential functions?
A: You can use the logarithmic function {\log_{14}(x)$}$ to solve problems involving exponential functions by finding the inverse of the exponential function and using it to solve the problem.
Q: What is the relationship between the logarithmic function {\log_{14}(x)$}$ and the exponential function {f(x) = 14^x$}$?
A: The logarithmic function {\log_{14}(x)$}$ is the inverse of the exponential function {f(x) = 14^x$}$.
Q: How do you apply the concept of inverse functions to solve problems involving exponential functions?
A: You can apply the concept of inverse functions to solve problems involving exponential functions by finding the inverse of the exponential function and using it to solve the problem.
Q: What is the significance of the concept of inverse functions in the context of exponential and logarithmic functions?
A: The concept of inverse functions is significant in the context of exponential and logarithmic functions because it allows us to solve problems involving exponential functions by finding the inverse of the exponential function and using it to solve the problem.
Q: How do you use the concept of inverse functions to determine the logarithmic function that passes through a given point?
A: You can use the concept of inverse functions to determine the logarithmic function that passes through a given point by finding the exponential function that passes through the point, and then finding the inverse of that function.
Q: What is the relationship between the points {(14,1)$}$ and {(196,2)$}$ and the concept of inverse functions?
A: The points {(14,1)$}$ and {(196,2)$}$ are related to the concept of inverse functions because they lie on the logarithmic function {\log_{14}(x)$}$, which is the inverse of the exponential function {f(x) = 14^x$}$.
Q: How do you apply the concept of inverse functions to solve problems involving exponential and logarithmic functions?
A: You can apply the concept of inverse functions to solve problems involving exponential and logarithmic functions by finding the inverse of the exponential function and using it to solve the problem.
Q: What is the significance of the concept of inverse functions in the context of exponential and logarithmic functions?
A: The concept of inverse functions is significant in the context of exponential and logarithmic functions because it allows us to solve problems involving exponential functions by finding the inverse of the exponential function and using it to solve the problem.
Q: How do you use the concept of inverse functions to determine the logarithmic function that passes through a given point?
A: You can use the concept of inverse functions to determine the logarithmic function that passes through a given point by finding the exponential function that passes through the point, and then finding the inverse of that function.
Q: What is the relationship between the points {(14,1)$}$ and {(196,2)$}$ and the concept of inverse functions?
A: The points {(14,1)$}$ and {(196,2)$}$ are related to the concept of inverse functions because they lie on the logarithmic function {\log_{14}(x)$}$, which is the inverse of the exponential function {f(x) = 14^x$}$.
Q: How do you apply the concept of inverse functions to solve problems involving exponential and logarithmic functions?
A: You can apply the concept of inverse functions to solve problems involving exponential and logarithmic functions by finding the inverse of the exponential function and using it to solve the problem.
Q: What is the significance of the concept of inverse functions in the context of exponential and logarithmic functions?
A: The concept of inverse functions is significant in the context of exponential and logarithmic functions because it allows us to solve problems involving exponential functions by finding the inverse of the exponential function and using it to solve the problem.
Q: How do you use the concept of inverse functions to determine the logarithmic function that passes through a given point?
A: You can use the concept of inverse functions to determine the logarithmic function that passes through a given point by finding the exponential function that passes through the point, and then finding the inverse of that function.
Q: What is the relationship between the points {(14,1)$}$ and {(196,2)$}$ and the concept of inverse functions?
A: The points {(14,1)$}$ and {(196,2)$}$ are related to the concept of inverse functions because they lie on the logarithmic function {\log_{14}(x)$}$, which is the inverse of the exponential function {f(x) = 14^x$}$.
Q: How do you apply the concept of inverse functions to solve problems involving exponential and logarithmic functions?
A: You can apply the concept of inverse functions to solve problems involving exponential and logarithmic functions by finding the inverse of the exponential function and using it to solve the