End Behavior of Logarithmic Functions

Oct 13, 2025 by miodeya.com 125 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their end behavior is crucial for solving various mathematical problems. In this article, we will explore the end behavior of the logarithmic function $f(x) = \log (x + 3) - 2$ and determine which statement about its end behavior is true.

The Logarithmic Function

The logarithmic function $f(x) = \log (x + 3) - 2$ is a transformation of the basic logarithmic function $f(x) = \log x$ . The transformation involves shifting the graph of the basic logarithmic function 3 units to the left and 2 units down.

End Behavior of Logarithmic Functions

The end behavior of a logarithmic function refers to the behavior of the function as $x$ approaches positive or negative infinity. For the basic logarithmic function $f(x) = \log x$ , the end behavior is as follows:

As $x$ approaches positive infinity, $y$ approaches positive infinity.
As $x$ approaches negative infinity, $y$ approaches negative infinity.

Vertical Asymptote

The vertical asymptote of a logarithmic function is the value of $x$ that makes the function undefined. In the case of the function $f(x) = \log (x + 3) - 2$ , the vertical asymptote is at $x = -3$ . This is because the function is undefined when $x + 3 = 0$ , which occurs when $x = -3$ .

End Behavior of the Given Function

Now, let's analyze the end behavior of the function $f(x) = \log (x + 3) - 2$ . As $x$ approaches positive infinity, $x + 3$ also approaches positive infinity, and therefore, $\log (x + 3)$ approaches positive infinity. Subtracting 2 from this value results in $y$ approaching negative infinity.

As $x$ approaches negative infinity, $x + 3$ approaches negative infinity, and therefore, $\log (x + 3)$ approaches negative infinity. Subtracting 2 from this value results in $y$ approaching negative infinity.

Statement Analysis

Now, let's analyze the given statements about the end behavior of the function $f(x) = \log (x + 3) - 2$ .

A. As $x$ decreases to the vertical asymptote at $x = -3$ , $y$ decreases to negative infinity.

B. As $x$ increases to the vertical asymptote at $x = -3$ , $y$ increases to positive infinity.

C. As $x$ approaches the vertical asymptote at $x = -3$ , $y$ approaches negative infinity.

D. As $x$ approaches the vertical asymptote at $x = -3$ , $y$ approaches positive infinity.

Conclusion

Based on the analysis of the end behavior of the function $f(x) = \log (x + 3) - 2$ , we can conclude that statement A is true. As $x$ decreases to the vertical asymptote at $x = -3$ , $y$ decreases to negative infinity.

Final Answer

The final answer is A.

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "End Behavior of Logarithmic Functions" by Purplemath
[3] "Vertical Asymptotes of Logarithmic Functions" by Mathway

Additional Resources

[1] "Logarithmic Functions" by Khan Academy
[2] "End Behavior of Logarithmic Functions" by IXL
[3] "Vertical Asymptotes of Logarithmic Functions" by Wolfram Alpha
Logarithmic Function End Behavior Q&A =====================================

Introduction

In our previous article, we explored the end behavior of the logarithmic function $f(x) = \log (x + 3) - 2$ . We analyzed the behavior of the function as $x$ approaches positive or negative infinity and determined that statement A is true. In this article, we will answer some frequently asked questions about the end behavior of logarithmic functions.

Q&A

Q: What is the end behavior of the basic logarithmic function $f(x) = \log x$ ?

A: The end behavior of the basic logarithmic function $f(x) = \log x$ is as follows:

As $x$ approaches positive infinity, $y$ approaches positive infinity.
As $x$ approaches negative infinity, $y$ approaches negative infinity.

Q: What is the vertical asymptote of the function $f(x) = \log (x + 3) - 2$ ?

A: The vertical asymptote of the function $f(x) = \log (x + 3) - 2$ is at $x = -3$ . This is because the function is undefined when $x + 3 = 0$ , which occurs when $x = -3$ .

Q: How does the end behavior of the function $f(x) = \log (x + 3) - 2$ change as $x$ approaches the vertical asymptote at $x = -3$ ?

A: As $x$ approaches the vertical asymptote at $x = -3$ , $y$ approaches negative infinity.

Q: What is the significance of the vertical asymptote in the context of logarithmic functions?

A: The vertical asymptote of a logarithmic function is the value of $x$ that makes the function undefined. It represents a point where the function is not defined and can be used to determine the behavior of the function as $x$ approaches that point.

Q: Can the end behavior of a logarithmic function be affected by transformations such as shifting or scaling?

A: Yes, the end behavior of a logarithmic function can be affected by transformations such as shifting or scaling. For example, if a logarithmic function is shifted to the left or right, its end behavior will change accordingly.

Q: How can the end behavior of a logarithmic function be used in real-world applications?

A: The end behavior of a logarithmic function can be used in various real-world applications, such as modeling population growth, chemical reactions, or financial transactions. Understanding the end behavior of a logarithmic function can help predict and analyze the behavior of complex systems.

Conclusion

In this article, we answered some frequently asked questions about the end behavior of logarithmic functions. We discussed the end behavior of the basic logarithmic function, the vertical asymptote of the function $f(x) = \log (x + 3) - 2$ , and the significance of the vertical asymptote in the context of logarithmic functions. We also explored how the end behavior of a logarithmic function can be affected by transformations and how it can be used in real-world applications.

Final Answer

The final answer is that the end behavior of a logarithmic function is a crucial concept in mathematics that can be used to analyze and predict the behavior of complex systems.

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "End Behavior of Logarithmic Functions" by Purplemath
[3] "Vertical Asymptotes of Logarithmic Functions" by Mathway

Additional Resources

[1] "Logarithmic Functions" by Khan Academy
[2] "End Behavior of Logarithmic Functions" by IXL
[3] "Vertical Asymptotes of Logarithmic Functions" by Wolfram Alpha