The Function F F F Is Logarithmic, And The Points { (2,1)$}$ And { (4,2)$}$ Are On The Graph Of F F F In The { Xy$}$-plane. Which Of The Following Could Define F ( X F(x F ( X ]?A. { \log_4 X$}$B.

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Introduction

In mathematics, logarithmic functions play a crucial role in various applications, including finance, science, and engineering. Given two points on the graph of a logarithmic function, we can determine the correct definition of the function. In this article, we will explore how to find the correct definition of a logarithmic function $f$ given two points on its graph.

Understanding Logarithmic Functions

A logarithmic function is a function that is the inverse of an exponential function. It is defined as:

$$f(x) = \log_a x$$

where $a$ is the base of the logarithm. The logarithmic function has a characteristic "S" shape, with the function increasing as $x$ increases.

Given Points on the Graph

We are given two points on the graph of $f$: $(2,1)$ and $(4,2)$. These points lie on the graph of $f$ in the $xy$-plane.

Using the Points to Find the Correct Definition

To find the correct definition of $f$, we can use the given points to set up equations. Since the points lie on the graph of $f$, we know that:

$$f(2) = 1$$

$$f(4) = 2$$

We can use these equations to find the correct definition of $f$.

Option A: $\log_4 x$

Let's consider option A: $\log_4 x$. We can plug in the given points to see if this definition satisfies the equations:

$$\log_4 2 = 1/2$$

$$\log_4 4 = 1$$

This definition does not satisfy the equations, so option A is not the correct definition of $f$.

Option B: $\log_2 x$

Let's consider option B: $\log_2 x$. We can plug in the given points to see if this definition satisfies the equations:

$$\log_2 2 = 1$$

$$\log_2 4 = 2$$

This definition satisfies the equations, so option B could define $f(x)$.

Conclusion

In this article, we explored how to find the correct definition of a logarithmic function $f$ given two points on its graph. We used the given points to set up equations and tested two possible definitions of $f$. We found that option B: $\log_2 x$ could define $f(x)$.

The Final Answer

The final answer is option B: $\log_2 x$.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Inverse Functions" by Khan Academy

Additional Resources

• [1] "Logarithmic Functions" by Wolfram MathWorld
• [2] "Inverse Functions" by MIT OpenCourseWare
  Logarithmic Functions: A Q&A Guide =====================================

Introduction

In our previous article, we explored how to find the correct definition of a logarithmic function $f$ given two points on its graph. In this article, we will answer some frequently asked questions about logarithmic functions.

Q: What is a logarithmic function?

A: A logarithmic function is a function that is the inverse of an exponential function. It is defined as:

$$f(x) = \log_a x$$

where $a$ is the base of the logarithm.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers. In other words, the input $x$ must be greater than 0.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function depends on the base of the logarithm. If the base is $a$, then the range is all real numbers.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use the following steps:

1. Determine the base of the logarithm.
2. Find the x-intercept by setting $y = 0$.
3. Find the y-intercept by setting $x = 1$.
4. Plot the points and draw a smooth curve through them.

Q: What is the relationship between logarithmic and exponential functions?

A: Logarithmic and exponential functions are inverse functions. This means that if $f(x) = \log_a x$, then $f^{-1}(x) = a^x$.

Q: How do I evaluate a logarithmic function?

A: To evaluate a logarithmic function, you can use the following steps:

1. Determine the base of the logarithm.
2. Use the definition of the logarithm to rewrite the function in exponential form.
3. Evaluate the exponential function.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

• $\log_2 x$
• $\log_3 x$
• $\log_10 x$
• $\log_e x$

Q: How do I use logarithmic functions in real-world applications?

A: Logarithmic functions have many real-world applications, including:

• Finance: Logarithmic functions are used to calculate interest rates and investment returns.
• Science: Logarithmic functions are used to model population growth and chemical reactions.
• Engineering: Logarithmic functions are used to design electronic circuits and calculate signal strengths.

Conclusion

In this article, we answered some frequently asked questions about logarithmic functions. We hope that this guide has been helpful in understanding logarithmic functions and their applications.

The Final Answer

There is no final answer to this article, as it is a Q&A guide.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Inverse Functions" by Khan Academy

Additional Resources

• [1] "Logarithmic Functions" by Wolfram MathWorld
• [2] "Inverse Functions" by MIT OpenCourseWare