The Domain and Range of Logarithmic and Exponential Functions

Oct 13, 2025 by windailys.com 22 views

The Domain and Range of Logarithmic and Exponential Functions

Understanding the Relationship Between Logarithmic and Exponential Functions

In mathematics, logarithmic and exponential functions are inverses of each other. This means that if we have a logarithmic function $f(x)=\ln x$ , its corresponding exponential function is $g(x)=e^x$ . The domain of the logarithmic function $f(x)=\ln x$ is $x\ \textgreater \ 0$ , which means that the input of the function must be a positive real number. In this article, we will explore the properties of the exponential function $g(x)=e^x$ and determine which statement is true about its range.

The Exponential Function $g(x)=e^x$

The exponential function $g(x)=e^x$ is a fundamental function in mathematics that has a wide range of applications in various fields, including physics, engineering, and economics. The function is defined as $g(x)=e^x$ , where $e$ is a mathematical constant approximately equal to 2.71828. The exponential function is an increasing function, which means that as the input $x$ increases, the output $g(x)$ also increases.

The Range of the Exponential Function $g(x)=e^x$

The range of a function is the set of all possible output values. In the case of the exponential function $g(x)=e^x$ , the range is all positive real numbers. This means that for any positive real number $y$ , there exists a real number $x$ such that $g(x)=y$ . In other words, the exponential function $g(x)=e^x$ takes on all positive real values.

Statement A: Function $g(x)$ has a range of $y\ \textgreater \ 0$

Based on our understanding of the exponential function $g(x)=e^x$ , we can conclude that statement A is true. The range of the exponential function $g(x)=e^x$ is indeed all positive real numbers, which can be represented as $y\ \textgreater \ 0$ .

Conclusion

In conclusion, the domain of the logarithmic function $f(x)=\ln x$ is $x\ \textgreater \ 0$ . The corresponding exponential function $g(x)=e^x$ has a range of all positive real numbers, which can be represented as $y\ \textgreater \ 0$ . This means that statement A is true about the range of the exponential function $g(x)=e^x$ .

Key Takeaways

The domain of the logarithmic function $f(x)=\ln x$ is $x\ \textgreater \ 0$ .
The corresponding exponential function $g(x)=e^x$ has a range of all positive real numbers.
The range of the exponential function $g(x)=e^x$ can be represented as $y\ \textgreater \ 0$ .

Frequently Asked Questions

What is the domain of the logarithmic function $f(x)=\ln x$ ?
The domain of the logarithmic function $f(x)=\ln x$ is $x\ \textgreater \ 0$ .
What is the range of the exponential function $g(x)=e^x$ ?
The range of the exponential function $g(x)=e^x$ is all positive real numbers, which can be represented as $y\ \textgreater \ 0$ .

References

[1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/expfunc.html
[2] "Logarithmic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/logfunc.html

Additional Resources

Khan Academy: Exponential Functions
Khan Academy: Logarithmic Functions
Wolfram MathWorld: Exponential Function
Wolfram MathWorld: Logarithmic Function
Q&A: Logarithmic and Exponential Functions

Understanding the Relationship Between Logarithmic and Exponential Functions

In our previous article, we explored the properties of logarithmic and exponential functions, including their domains and ranges. In this article, we will answer some frequently asked questions about logarithmic and exponential functions.

Q: What is the domain of the logarithmic function $f(x)=\ln x$ ?

A: The domain of the logarithmic function $f(x)=\ln x$ is $x\ \textgreater \ 0$ . This means that the input of the function must be a positive real number.

Q: What is the range of the exponential function $g(x)=e^x$ ?

A: The range of the exponential function $g(x)=e^x$ is all positive real numbers, which can be represented as $y\ \textgreater \ 0$ . This means that for any positive real number $y$ , there exists a real number $x$ such that $g(x)=y$ .

Q: How do logarithmic and exponential functions relate to each other?

A: Logarithmic and exponential functions are inverses of each other. This means that if we have a logarithmic function $f(x)=\ln x$ , its corresponding exponential function is $g(x)=e^x$ . Similarly, if we have an exponential function $g(x)=e^x$ , its corresponding logarithmic function is $f(x)=\ln x$ .

Q: What is the relationship between the logarithmic function $f(x)=\ln x$ and the exponential function $g(x)=e^x$ ?

A: The logarithmic function $f(x)=\ln x$ and the exponential function $g(x)=e^x$ are inverses of each other. This means that if we have a logarithmic function $f(x)=\ln x$ , its corresponding exponential function is $g(x)=e^x$ , and vice versa.

Q: How do we use logarithmic and exponential functions in real-world applications?

A: Logarithmic and exponential functions have a wide range of applications in various fields, including physics, engineering, and economics. For example, we can use logarithmic functions to model population growth, and exponential functions to model compound interest.

Q: What are some common mistakes to avoid when working with logarithmic and exponential functions?

A: Some common mistakes to avoid when working with logarithmic and exponential functions include:

Not checking the domain and range of the function
Not using the correct inverse function
Not simplifying the expression
Not checking for extraneous solutions

Q: How do we simplify logarithmic and exponential expressions?

A: To simplify logarithmic and exponential expressions, we can use the following properties:

Logarithmic properties:

$\ln (ab) = \ln a + \ln b$
$\ln (a/b) = \ln a - \ln b$
$\ln (a^b) = b\ln a$

Exponential properties:

$e^{\ln x} = x$
$\ln e^x = x$

Q: What are some common applications of logarithmic and exponential functions?

A: Some common applications of logarithmic and exponential functions include:

Modeling population growth
Modeling compound interest
Modeling chemical reactions
Modeling electrical circuits