Solve The Equation For $x$. Give An Exact Answer And A Four-decimal-place Approximation.$\ln X = 2.4$The Exact Answer Is $x =$ $\square$ (Simplify Your Answer.)A Four-decimal-place Approximation Is $x =$

Introduction

In this article, we will delve into solving the equation $\ln x = 2.4$ to find the exact value of $x$ and a four-decimal-place approximation. The natural logarithm function, denoted by $\ln x$, is the inverse of the exponential function $e^x$. It is a fundamental concept in mathematics, particularly in calculus and algebra. Understanding how to solve equations involving logarithms is crucial for solving various mathematical problems.

Understanding the Natural Logarithm Function

The natural logarithm function, $\ln x$, is defined as the power to which the base $e$ must be raised to produce the number $x$. In other words, if $y = \ln x$, then $e^y = x$. This function has several important properties, including:

• The domain of $\ln x$ is all positive real numbers.
• The range of $\ln x$ is all real numbers.
• The graph of $\ln x$ is a continuous, increasing function.

Solving the Equation $\ln x = 2.4$

To solve the equation $\ln x = 2.4$, we need to isolate $x$ by applying the exponential function to both sides of the equation. This is because the exponential function is the inverse of the natural logarithm function.

$$\ln x = 2.4$$

Applying the exponential function to both sides:

$$e^{\ln x} = e^{2.4}$$

Using the property that $e^{\ln x} = x$, we get:

$$x = e^{2.4}$$

Finding the Exact Value of x

To find the exact value of $x$, we need to evaluate the expression $e^{2.4}$. This can be done using a calculator or by using the properties of exponents.

$$x = e^{2.4}$$

Using a calculator, we get:

$$x ≈ 11.021$$

However, we are asked to find the exact value of $x$. Since $e^{2.4}$ is an irrational number, we cannot simplify it further. Therefore, the exact value of $x$ is:

$$x = e^{2.4}$$

Finding a Four-Decimal-Place Approximation of x

To find a four-decimal-place approximation of $x$, we can use a calculator to evaluate the expression $e^{2.4}$.

$$x ≈ 11.021$$

Rounding this value to four decimal places, we get:

$$x ≈ 11.0210$$

Conclusion

In this article, we solved the equation $\ln x = 2.4$ to find the exact value of $x$ and a four-decimal-place approximation. We used the properties of the natural logarithm function and the exponential function to isolate $x$ and evaluate the expression $e^{2.4}$. The exact value of $x$ is $e^{2.4}$, and a four-decimal-place approximation is $11.0210$. Understanding how to solve equations involving logarithms is crucial for solving various mathematical problems, and this article provides a step-by-step guide on how to do so.

Frequently Asked Questions

• What is the natural logarithm function?
• How do you solve equations involving logarithms?
• What is the exact value of $x$ in the equation $\ln x = 2.4$?
• What is a four-decimal-place approximation of $x$ in the equation $\ln x = 2.4$?

Final Answer

The final answer is: $\boxed{e^{2.4}}$

Introduction

In our previous article, we solved the equation $\ln x = 2.4$ to find the exact value of $x$ and a four-decimal-place approximation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving equations involving logarithms.

Q&A

Q1: What is the natural logarithm function?

A1: The natural logarithm function, denoted by $\ln x$, is the inverse of the exponential function $e^x$. It is a fundamental concept in mathematics, particularly in calculus and algebra.

Q2: How do you solve equations involving logarithms?

A2: To solve equations involving logarithms, you need to isolate the logarithmic term and then apply the exponential function to both sides of the equation. This is because the exponential function is the inverse of the natural logarithm function.

Q3: What is the exact value of $x$ in the equation $\ln x = 2.4$?

A3: The exact value of $x$ in the equation $\ln x = 2.4$ is $e^{2.4}$. This is because the exponential function is the inverse of the natural logarithm function.

Q4: What is a four-decimal-place approximation of $x$ in the equation $\ln x = 2.4$?

A4: A four-decimal-place approximation of $x$ in the equation $\ln x = 2.4$ is $11.0210$. This is obtained by evaluating the expression $e^{2.4}$ using a calculator and rounding the result to four decimal places.

Q5: How do you evaluate the expression $e^{2.4}$?

A5: To evaluate the expression $e^{2.4}$, you can use a calculator or apply the properties of exponents. Using a calculator, you get $e^{2.4} ≈ 11.021$. This value can be rounded to four decimal places to obtain $11.0210$.

Q6: What is the domain of the natural logarithm function?

A6: The domain of the natural logarithm function is all positive real numbers. This means that the input of the function must be a positive number.

Q7: What is the range of the natural logarithm function?

A7: The range of the natural logarithm function is all real numbers. This means that the output of the function can be any real number.

Q8: How do you graph the natural logarithm function?

A8: To graph the natural logarithm function, you can use a graphing calculator or a computer algebra system. The graph of the function is a continuous, increasing function.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving equations involving logarithms. We covered topics such as the natural logarithm function, solving equations involving logarithms, and evaluating expressions involving exponents. We hope that this guide has been helpful in clarifying any doubts you may have had.

Frequently Asked Questions

• What is the natural logarithm function?
• How do you solve equations involving logarithms?
• What is the exact value of $x$ in the equation $\ln x = 2.4$?
• What is a four-decimal-place approximation of $x$ in the equation $\ln x = 2.4$?
• How do you evaluate the expression $e^{2.4}$?
• What is the domain of the natural logarithm function?
• What is the range of the natural logarithm function?
• How do you graph the natural logarithm function?

Final Answer

The final answer is: $\boxed{e^{2.4}}$