Solve Ln ⁡ ( 2 X − 1 ) = 8 \ln (2x - 1) = 8 Ln ( 2 X − 1 ) = 8 . Round To The Nearest Thousandth.A. 2981.458 B. 1489.979 C. 2979.958 D. 1490.979

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will focus on solving a specific logarithmic equation, $\ln (2x - 1) = 8$, and provide a step-by-step guide on how to approach it.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions. In this case, we are dealing with a natural logarithm, which is the logarithm to the base $e$.

The Equation

The equation we are given is $\ln (2x - 1) = 8$. This equation involves a natural logarithm, and our goal is to solve for the variable $x$.

Step 1: Exponentiate Both Sides

To solve this equation, we need to get rid of the logarithm. We can do this by exponentiating both sides of the equation. Since the logarithm is a natural logarithm, we will use the exponential function $e^x$ to exponentiate both sides.

$$\ln (2x - 1) = 8$$

$$e^{\ln (2x - 1)} = e^8$$

Using the property of logarithms that states $e^{\ln x} = x$, we can simplify the left-hand side of the equation.

$$2x - 1 = e^8$$

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation is $e^8$. We can simplify this expression by using a calculator or a computer program to find the value of $e^8$.

$$e^8 \approx 2980.957$$

Step 3: Solve for x

Now that we have simplified the right-hand side of the equation, we can solve for $x$.

$$2x - 1 = 2980.957$$

$$2x = 2980.957 + 1$$

$$2x = 2981.957$$

$$x = \frac{2981.957}{2}$$

$$x \approx 1490.9785$$

Rounding to the Nearest Thousandth

We are asked to round the solution to the nearest thousandth. To do this, we need to round the value of $x$ to three decimal places.

$$x \approx 1490.979$$

Conclusion

In this article, we have solved the logarithmic equation $\ln (2x - 1) = 8$ using a step-by-step approach. We have exponentiated both sides of the equation, simplified the right-hand side, and solved for $x$. The solution is $x \approx 1490.979$, which is the correct answer.

Answer

The correct answer is:

• B. 1489.979
  Solving Logarithmic Equations: A Q&A Guide =====================================================

Introduction

In our previous article, we solved the logarithmic equation $\ln (2x - 1) = 8$ using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation involves a logarithmic function, while an exponential equation involves an exponential function. For example, the equation $\ln (2x - 1) = 8$ is a logarithmic equation, while the equation $2^x = 8$ is an exponential equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to follow these steps:

1. Exponentiate both sides of the equation using the exponential function.
2. Simplify the right-hand side of the equation.
3. Solve for the variable.

Q: What is the property of logarithms that states $e^{\ln x} = x$?

A: This property is known as the inverse property of logarithms. It states that the exponential function and the natural logarithm are inverse functions, and that $e^{\ln x} = x$ for all positive real numbers $x$.

Q: How do I use a calculator or computer program to find the value of $e^8$?

A: You can use a calculator or computer program to find the value of $e^8$ by entering the expression $e^8$ and pressing the "calculate" or "evaluate" button. Alternatively, you can use a mathematical software package such as Mathematica or Maple to find the value of $e^8$.

Q: What is the solution to the equation $\ln (2x - 1) = 8$?

A: The solution to the equation $\ln (2x - 1) = 8$ is $x \approx 1490.979$.

Q: How do I round the solution to the nearest thousandth?

A: To round the solution to the nearest thousandth, you need to round the value of $x$ to three decimal places. In this case, the solution is $x \approx 1490.979$, which is already rounded to three decimal places.

Q: What is the correct answer to the equation $\ln (2x - 1) = 8$?

A: The correct answer to the equation $\ln (2x - 1) = 8$ is:

• B. 1489.979

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques involved in solving logarithmic equations. We have answered common questions about logarithmic equations, including how to solve them, what the property of logarithms is, and how to use a calculator or computer program to find the value of $e^8$. We hope that this guide has been helpful in understanding logarithmic equations.

Additional Resources

If you are interested in learning more about logarithmic equations, we recommend the following resources:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Answer

The correct answer is:

• B. 1489.979