What Is The Value Of E Ln ⁡ 7 X E^{\ln 7 X} E L N 7 X ?A. 1 B. 7 E 7 E 7 E C. 7 X 7 X 7 X D. 7

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Introduction

In mathematics, the natural exponential function and the natural logarithm function are two fundamental functions that are closely related. The natural exponential function, denoted by $e^x$, is the inverse of the natural logarithm function, denoted by $\ln x$. In this article, we will explore the value of the expression $e^{\ln 7 x}$ and provide a step-by-step solution to this problem.

Understanding the Natural Exponential Function

The natural exponential function, denoted by $e^x$, is a mathematical function that takes a real number $x$ as input and returns a positive real number as output. The function is defined as:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

where $n!$ denotes the factorial of $n$. The natural exponential function has several important properties, including:

• Domain: The domain of the natural exponential function is all real numbers, denoted by $\mathbb{R}$.
• Range: The range of the natural exponential function is all positive real numbers, denoted by $(0, \infty)$.
• Inverse: The natural exponential function is the inverse of the natural logarithm function, denoted by $\ln x$.

Understanding the Natural Logarithm Function

The natural logarithm function, denoted by $\ln x$, is a mathematical function that takes a positive real number $x$ as input and returns a real number as output. The function is defined as:

$$\ln x = \int_{1}^{x} \frac{1}{t} dt$$

The natural logarithm function has several important properties, including:

• Domain: The domain of the natural logarithm function is all positive real numbers, denoted by $(0, \infty)$.
• Range: The range of the natural logarithm function is all real numbers, denoted by $\mathbb{R}$.
• Inverse: The natural logarithm function is the inverse of the natural exponential function, denoted by $e^x$.

Evaluating the Expression $e^{\ln 7 x}$

Now that we have a good understanding of the natural exponential function and the natural logarithm function, we can evaluate the expression $e^{\ln 7 x}$. Using the property of the natural exponential function being the inverse of the natural logarithm function, we can rewrite the expression as:

$$e^{\ln 7 x} = e^{\ln (7 x)}$$

Using the property of the natural logarithm function being the inverse of the natural exponential function, we can rewrite the expression as:

$$e^{\ln (7 x)} = 7 x$$

Therefore, the value of the expression $e^{\ln 7 x}$ is $7 x$.

Conclusion

In this article, we have explored the value of the expression $e^{\ln 7 x}$ and provided a step-by-step solution to this problem. We have used the properties of the natural exponential function and the natural logarithm function to evaluate the expression. The final answer is $7 x$.

Answer Key

The correct answer is C. $7 x$.

Discussion

This problem is a great example of how the natural exponential function and the natural logarithm function are related. The property of the natural exponential function being the inverse of the natural logarithm function is a fundamental concept in mathematics, and it is used extensively in many areas of mathematics, including calculus, differential equations, and probability theory.

Additional Resources

For more information on the natural exponential function and the natural logarithm function, please refer to the following resources:

• Wikipedia: Natural Exponential Function and Natural Logarithm Function
• MathWorld: Natural Exponential Function and Natural Logarithm Function
• Khan Academy: Natural Exponential Function and Natural Logarithm Function

References

• Spivak, M. (1965). Calculus. W.A. Benjamin.
• Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• Bartle, R. G. (1976). The Elements of Real Analysis. John Wiley & Sons.
  Q&A: What is the Value of $e^{\ln 7 x}$? =====================================================

Introduction

In our previous article, we explored the value of the expression $e^{\ln 7 x}$ and provided a step-by-step solution to this problem. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the natural exponential function?

A: The natural exponential function, denoted by $e^x$, is a mathematical function that takes a real number $x$ as input and returns a positive real number as output. The function is defined as:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Q: What is the natural logarithm function?

A: The natural logarithm function, denoted by $\ln x$, is a mathematical function that takes a positive real number $x$ as input and returns a real number as output. The function is defined as:

$$\ln x = \int_{1}^{x} \frac{1}{t} dt$$

Q: How are the natural exponential function and the natural logarithm function related?

A: The natural exponential function and the natural logarithm function are inverses of each other. This means that if we take the natural logarithm of a number and then take the natural exponential of the result, we get back the original number.

Q: What is the value of $e^{\ln 7 x}$?

A: Using the property of the natural exponential function being the inverse of the natural logarithm function, we can rewrite the expression as:

$$e^{\ln 7 x} = e^{\ln (7 x)}$$

Using the property of the natural logarithm function being the inverse of the natural exponential function, we can rewrite the expression as:

$$e^{\ln (7 x)} = 7 x$$

Therefore, the value of the expression $e^{\ln 7 x}$ is $7 x$.

Q: Can you provide more examples of how to evaluate expressions involving the natural exponential function and the natural logarithm function?

A: Here are a few more examples:

• $e^{\ln 2} = 2$
• $e^{\ln (3 x)} = 3 x$
• $\ln (e^x) = x$

Q: What are some common applications of the natural exponential function and the natural logarithm function?

A: The natural exponential function and the natural logarithm function have many applications in mathematics, science, and engineering. Some common applications include:

• Calculus: The natural exponential function and the natural logarithm function are used extensively in calculus, including in the study of limits, derivatives, and integrals.
• Differential equations: The natural exponential function and the natural logarithm function are used to solve differential equations, which are used to model a wide range of phenomena in physics, engineering, and other fields.
• Probability theory: The natural exponential function and the natural logarithm function are used in probability theory to model random variables and to calculate probabilities.

Conclusion

In this article, we have answered some frequently asked questions related to the value of the expression $e^{\ln 7 x}$. We have also provided some additional examples and applications of the natural exponential function and the natural logarithm function. We hope that this article has been helpful in clarifying some of the concepts and ideas related to this topic.

Answer Key

The correct answers to the questions are:

• Q1: The natural exponential function is a mathematical function that takes a real number $x$ as input and returns a positive real number as output.
• Q2: The natural logarithm function is a mathematical function that takes a positive real number $x$ as input and returns a real number as output.
• Q3: The natural exponential function and the natural logarithm function are inverses of each other.
• Q4: The value of $e^{\ln 7 x}$ is $7 x$.
• Q5: Yes, here are a few more examples:

• $e^{\ln 2} = 2$
• $e^{\ln (3 x)} = 3 x$
• $\ln (e^x) = x$

• Q6: Some common applications of the natural exponential function and the natural logarithm function include calculus, differential equations, and probability theory.

Discussion

This article is a great resource for anyone who wants to learn more about the natural exponential function and the natural logarithm function. We hope that this article has been helpful in clarifying some of the concepts and ideas related to this topic.

Additional Resources

For more information on the natural exponential function and the natural logarithm function, please refer to the following resources:

• Wikipedia: Natural Exponential Function and Natural Logarithm Function
• MathWorld: Natural Exponential Function and Natural Logarithm Function
• Khan Academy: Natural Exponential Function and Natural Logarithm Function

References

• Spivak, M. (1965). Calculus. W.A. Benjamin.
• Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• Bartle, R. G. (1976). The Elements of Real Analysis. John Wiley & Sons.