Evaluate The Following Expression Without Using A Calculator:$e^{\ln 5}$A. 0 B. 10 C. 1 D. 5

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Introduction

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In this article, we will evaluate the expression $e^{\ln 5}$ without using a calculator. This expression involves the natural exponential function and the natural logarithm function. We will use the properties of these functions to simplify the expression and find its value.

Understanding the Natural Exponential Function

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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 natural exponential 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:

• Exponential property: $e^{x+y} = e^x \cdot e^y$
• Inverse property: $e^{\ln x} = x$ for $x > 0$
• Derivative property: $\frac{d}{dx} e^x = e^x$

Understanding the Natural Logarithm Function

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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 natural logarithm function is defined as:

$$\ln x = \int_1^x \frac{1}{t} dt$$

The natural logarithm function has several important properties, including:

• Logarithmic property: $\ln (xy) = \ln x + \ln y$
• Inverse property: $\ln e^x = x$ for $x \in \mathbb{R}$
• Derivative property: $\frac{d}{dx} \ln x = \frac{1}{x}$

Evaluating the Expression $e^{\ln 5}$

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Using the properties of the natural exponential function and the natural logarithm function, we can simplify the expression $e^{\ln 5}$ as follows:

$$e^{\ln 5} = e^{\ln 5} \cdot e^0 = e^{\ln 5} \cdot 1 = e^{\ln 5}$$

Since $e^{\ln x} = x$ for $x > 0$, we have:

$$e^{\ln 5} = 5$$

Therefore, the value of the expression $e^{\ln 5}$ is 5.

Conclusion

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In this article, we evaluated the expression $e^{\ln 5}$ without using a calculator. We used the properties of the natural exponential function and the natural logarithm function to simplify the expression and find its value. The final answer is 5.

Frequently Asked Questions

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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.

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.

Q: How do you evaluate the expression $e^{\ln 5}$?

A: Using the properties of the natural exponential function and the natural logarithm function, we can simplify the expression $e^{\ln 5}$ as follows: $e^{\ln 5} = e^{\ln 5} \cdot e^0 = e^{\ln 5} \cdot 1 = e^{\ln 5}$. Since $e^{\ln x} = x$ for $x > 0$, we have: $e^{\ln 5} = 5$.

Q: What is the final answer to the expression $e^{\ln 5}$?

A: The final answer to the expression $e^{\ln 5}$ is 5.

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Introduction

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In our previous article, we evaluated the expression $e^{\ln 5}$ without using a calculator. We used the properties of the natural exponential function and the natural logarithm function to simplify the expression and find its value. In this article, we will answer some frequently asked questions related to the expression $e^{\ln 5}$.

Q&A

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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.

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.

Q: How do you evaluate the expression $e^{\ln 5}$?

A: Using the properties of the natural exponential function and the natural logarithm function, we can simplify the expression $e^{\ln 5}$ as follows: $e^{\ln 5} = e^{\ln 5} \cdot e^0 = e^{\ln 5} \cdot 1 = e^{\ln 5}$. Since $e^{\ln x} = x$ for $x > 0$, we have: $e^{\ln 5} = 5$.

Q: What is the final answer to the expression $e^{\ln 5}$?

A: The final answer to the expression $e^{\ln 5}$ is 5.

Q: Can you explain the properties of the natural exponential function?

A: Yes, the natural exponential function has several important properties, including:

• Exponential property: $e^{x+y} = e^x \cdot e^y$
• Inverse property: $e^{\ln x} = x$ for $x > 0$
• Derivative property: $\frac{d}{dx} e^x = e^x$

Q: Can you explain the properties of the natural logarithm function?

A: Yes, the natural logarithm function has several important properties, including:

• Logarithmic property: $\ln (xy) = \ln x + \ln y$
• Inverse property: $\ln e^x = x$ for $x \in \mathbb{R}$
• Derivative property: $\frac{d}{dx} \ln x = \frac{1}{x}$

Q: How do you use the properties of the natural exponential function and the natural logarithm function to simplify the expression $e^{\ln 5}$?

A: To simplify the expression $e^{\ln 5}$, we can use the properties of the natural exponential function and the natural logarithm function as follows:

• Exponential property: $e^{\ln 5} = e^{\ln 5} \cdot e^0 = e^{\ln 5} \cdot 1 = e^{\ln 5}$
• Inverse property: $e^{\ln 5} = 5$

Q: What is the significance of the expression $e^{\ln 5}$?

A: The expression $e^{\ln 5}$ is significant because it demonstrates the relationship between the natural exponential function and the natural logarithm function. The expression $e^{\ln 5}$ is equal to 5, which shows that the natural exponential function and the natural logarithm function are inverse functions.

Conclusion

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In this article, we answered some frequently asked questions related to the expression $e^{\ln 5}$. We explained the properties of the natural exponential function and the natural logarithm function, and we demonstrated how to use these properties to simplify the expression $e^{\ln 5}$. The final answer to the expression $e^{\ln 5}$ is 5.

Frequently Asked Questions

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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.

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.

Q: How do you evaluate the expression $e^{\ln 5}$?

A: Using the properties of the natural exponential function and the natural logarithm function, we can simplify the expression $e^{\ln 5}$ as follows: $e^{\ln 5} = e^{\ln 5} \cdot e^0 = e^{\ln 5} \cdot 1 = e^{\ln 5}$. Since $e^{\ln x} = x$ for $x > 0$, we have: $e^{\ln 5} = 5$.

Q: What is the final answer to the expression $e^{\ln 5}$?

A: The final answer to the expression $e^{\ln 5}$ is 5.

Additional Resources

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• Natural Exponential Function
• Natural Logarithm Function
• Inverse Functions