What Is The Solution To The Equation Below? Round Your Answer To Two Decimal Places.$\ln X = -2$A. $x = 0.34$ B. $x = 0.14$ C. $x = -0.69$ D. $x = -7.39$

Feb 27, 2025 by ADMIN 159 views

Solving the Equation: Unraveling the Mystery of Natural Logarithms

In the realm of mathematics, equations often pose a challenge to solve, and the equation $\ln x = -2$ is no exception. The presence of the natural logarithm function, denoted by $\ln$ , adds a layer of complexity to the equation. In this article, we will delve into the world of logarithms and explore the solution to the given equation. We will also examine the answer choices and determine which one is correct.

Understanding the Natural Logarithm Function

The natural logarithm function, denoted by $\ln$ , is a fundamental concept in mathematics. It is defined as the inverse of the exponential function, $e^x$ . In other words, $\ln x$ is the power to which the base $e$ must be raised to obtain the number $x$ . The natural logarithm function has several properties that make it a powerful tool in mathematics.

Properties of the Natural Logarithm Function

The natural logarithm function has several properties that are essential to understand when working with logarithmic equations. Some of the key properties include:

Domain and Range: The domain of the natural logarithm function is all positive real numbers, while the range is all real numbers.
Inverse Function: The natural logarithm function is the inverse of the exponential function, $e^x$ .
Logarithmic Identity: $\ln e^x = x$ for all real numbers $x$ .
Power Rule: $\ln x^a = a \ln x$ for all real numbers $x$ and $a$ .

Solving the Equation

Now that we have a good understanding of the natural logarithm function, we can proceed to solve the equation $\ln x = -2$ . To solve this equation, we can use the property of the natural logarithm function that states $\ln e^x = x$ for all real numbers $x$ . We can rewrite the equation as:

\ln x = -2

e^{\ln x} = e^{-2}

x = e^{-2}

Evaluating the Answer Choices

Now that we have the solution to the equation, we can evaluate the answer choices. The answer choices are:

A. $x = 0.34$ B. $x = 0.14$ C. $x = -0.69$ D. $x = -7.39$

We can compare the solution we obtained, $x = e^{-2}$ , with the answer choices. Using a calculator, we can evaluate the value of $e^{-2}$ .

e^{-2} \approx 0.1353

Comparing this value with the answer choices, we can see that the closest answer choice is:

B. $x = 0.14$

In this article, we solved the equation $\ln x = -2$ using the properties of the natural logarithm function. We obtained the solution $x = e^{-2}$ and evaluated the answer choices. The closest answer choice is B. $x = 0.14$ . We hope this article has provided a clear understanding of the solution to the equation and the properties of the natural logarithm function.

The final answer is B. $x = 0.14$ .
Frequently Asked Questions: Natural Logarithms and Equations

In our previous article, we explored the solution to the equation $\ln x = -2$ using the properties of the natural logarithm function. In this article, we will address some of the most frequently asked questions related to natural logarithms and equations.

Q: What is the natural logarithm function?

A: The natural logarithm function, denoted by $\ln$ , is a fundamental concept in mathematics. It is defined as the inverse of the exponential function, $e^x$ . In other words, $\ln x$ is the power to which the base $e$ must be raised to obtain the number $x$ .

Q: What are the properties of the natural logarithm function?

A: The natural logarithm function has several properties that are essential to understand when working with logarithmic equations. Some of the key properties include:

Domain and Range: The domain of the natural logarithm function is all positive real numbers, while the range is all real numbers.
Inverse Function: The natural logarithm function is the inverse of the exponential function, $e^x$ .
Logarithmic Identity: $\ln e^x = x$ for all real numbers $x$ .
Power Rule: $\ln x^a = a \ln x$ for all real numbers $x$ and $a$ .

Q: How do I solve an equation with a natural logarithm?

A: To solve an equation with a natural logarithm, you can use the properties of the natural logarithm function. One of the most common methods is to rewrite the equation in exponential form using the property $\ln e^x = x$ .

Q: What is the difference between the natural logarithm and the common logarithm?

A: The natural logarithm, denoted by $\ln$ , is the inverse of the exponential function, $e^x$ . The common logarithm, denoted by $\log$ , is the inverse of the base-10 exponential function, $10^x$ . While both logarithms are used to solve equations, the natural logarithm is more commonly used in mathematics and science.

Q: How do I evaluate the natural logarithm of a number?

A: To evaluate the natural logarithm of a number, you can use a calculator or a computer program. Most calculators have a built-in function for the natural logarithm, denoted by $\ln$ . You can also use a computer program, such as Python or MATLAB, to evaluate the natural logarithm.

Q: What are some common applications of natural logarithms?

A: Natural logarithms have many applications in mathematics, science, and engineering. Some common applications include:

Finance: Natural logarithms are used to calculate the return on investment and the growth rate of an investment.
Biology: Natural logarithms are used to model population growth and the spread of diseases.
Physics: Natural logarithms are used to describe the behavior of particles in a gas and the decay of radioactive materials.

In this article, we addressed some of the most frequently asked questions related to natural logarithms and equations. We hope this article has provided a clear understanding of the natural logarithm function and its properties. If you have any further questions, please don't hesitate to ask.

The final answer is B. $x = 0.14$ .