End Behavior of the Logarithmic Function

Oct 13, 2025 by miodeya.com 35 views

Introduction

In mathematics, the end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity. Understanding the end behavior of a function is crucial in various mathematical and real-world applications. In this article, we will explore the end behavior of the logarithmic function $f(x) = \log(x - 2)$ and determine which statement describes its end behavior.

The Logarithmic Function

The logarithmic function $f(x) = \log(x - 2)$ is a type of function that is defined only for positive real numbers. The function is a composition of the logarithmic function and the linear function $g(x) = x - 2$ . The logarithmic function is a continuous and differentiable function that has a number of interesting properties.

End Behavior of the Logarithmic Function

To determine the end behavior of the logarithmic function, we need to consider the behavior of the function as the input values approach positive or negative infinity. As the input values approach positive infinity, the logarithmic function approaches negative infinity. This is because the logarithmic function is a decreasing function, and as the input values increase, the output values decrease.

Statement A: As the value of $x$ decreases, the value of $f(x)$ moves toward positive infinity

This statement is incorrect. As the value of $x$ decreases, the value of $f(x)$ does not move toward positive infinity. Instead, the value of $f(x)$ approaches negative infinity as the value of $x$ decreases.

Statement B: As the value of $x$ increases, the value of $f(x)$ moves toward negative infinity

This statement is correct. As the value of $x$ increases, the value of $f(x)$ approaches negative infinity. This is because the logarithmic function is a decreasing function, and as the input values increase, the output values decrease.

Conclusion

In conclusion, the end behavior of the logarithmic function $f(x) = \log(x - 2)$ is that as the value of $x$ increases, the value of $f(x)$ moves toward negative infinity. This is because the logarithmic function is a decreasing function, and as the input values increase, the output values decrease.

Key Takeaways

The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity.
The logarithmic function $f(x) = \log(x - 2)$ is a decreasing function that approaches negative infinity as the input values increase.
The correct statement that describes the end behavior of the logarithmic function is: As the value of $x$ increases, the value of $f(x)$ moves toward negative infinity.

Additional Resources

For more information on the end behavior of functions, please refer to the following resources:

Khan Academy: End Behavior of Functions
Mathway: End Behavior of Functions
Wolfram MathWorld: End Behavior of Functions

Final Thoughts

$Understanding the end behavior of a function is crucial in various mathematical and real-world applications. In this article, we explored the end behavior of the logarithmic function $f(x) = \log(x - 2)$ and determined which statement describes its end behavior. We hope that this article has provided valuable insights into the end behavior of functions and has helped you to better understand this important mathematical concept.<br/> **Q&A: Understanding the End Behavior of a Logarithmic Function**$

Introduction

In our previous article, we explored the end behavior of the logarithmic function $f(x) = \log(x - 2)$ and determined which statement describes its end behavior. In this article, we will answer some frequently asked questions about the end behavior of a logarithmic function.

Q: What is the end behavior of a logarithmic function?

A: The end behavior of a logarithmic function refers to the behavior of the function as the input values approach positive or negative infinity. In the case of the logarithmic function $f(x) = \log(x - 2)$ , the end behavior is that as the value of $x$ increases, the value of $f(x)$ approaches negative infinity.

Q: Why does the logarithmic function approach negative infinity as the input values increase?

A: The logarithmic function approaches negative infinity as the input values increase because the logarithmic function is a decreasing function. As the input values increase, the output values decrease.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers. In the case of the logarithmic function $f(x) = \log(x - 2)$ , the domain is all real numbers greater than 2.

Q: Can a logarithmic function have a horizontal asymptote?

A: No, a logarithmic function cannot have a horizontal asymptote. As the input values approach positive or negative infinity, the output values approach negative infinity.

Q: How does the end behavior of a logarithmic function compare to the end behavior of a polynomial function?

A: The end behavior of a logarithmic function is different from the end behavior of a polynomial function. A polynomial function can have a horizontal asymptote, while a logarithmic function approaches negative infinity as the input values increase.

Q: Can the end behavior of a logarithmic function be determined using the first derivative of the function?

A: Yes, the end behavior of a logarithmic function can be determined using the first derivative of the function. If the first derivative of the function is negative, then the function is decreasing and approaches negative infinity as the input values increase.

Q: What is the significance of the end behavior of a logarithmic function in real-world applications?

A: The end behavior of a logarithmic function is significant in real-world applications because it can be used to model population growth, chemical reactions, and other phenomena that involve exponential growth or decay.

Conclusion

In conclusion, the end behavior of a logarithmic function is an important concept in mathematics that has significant implications in real-world applications. We hope that this article has provided valuable insights into the end behavior of a logarithmic function and has helped you to better understand this important mathematical concept.

Additional Resources

For more information on the end behavior of functions, please refer to the following resources:

Khan Academy: End Behavior of Functions
Mathway: End Behavior of Functions
Wolfram MathWorld: End Behavior of Functions

Final Thoughts

Understanding the end behavior of a logarithmic function is crucial in various mathematical and real-world applications. We hope that this article has provided valuable insights into the end behavior of a logarithmic function and has helped you to better understand this important mathematical concept.