Solve The Logarithmic Equation Algebraically. Approximate The Result To Three Decimal Places. (If There Is No Solution, Enter NO SOLUTION.) Ln ⁡ X − Ln ⁡ ( X + 8 ) = 9 \ln X - \ln (x+8) = 9 Ln X − Ln ( X + 8 ) = 9 X = X = \, \, \, \, \, X =

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them algebraically is a crucial skill for students and professionals alike. In this article, we will focus on solving the logarithmic equation $\ln x - \ln (x+8) = 9$ algebraically and approximating the result to three decimal places.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's briefly review the concept of logarithmic equations. A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The logarithmic function is defined as $\log_b a = c$, where $b$ is the base of the logarithm, $a$ is the argument of the logarithm, and $c$ is the result of the logarithm.

Solving the Logarithmic Equation

To solve the logarithmic equation $\ln x - \ln (x+8) = 9$, we can use the properties of logarithms. Specifically, we can use the property that $\ln a - \ln b = \ln \frac{a}{b}$.

Using this property, we can rewrite the equation as:

$$\ln \frac{x}{x+8} = 9$$

Using Exponential Functions to Solve the Equation

To solve the equation, we can use the exponential function to eliminate the logarithm. Specifically, we can raise both sides of the equation to the power of $e$ (the base of the natural logarithm).

$$e^{\ln \frac{x}{x+8}} = e^9$$

Using the property that $e^{\ln a} = a$, we can simplify the equation to:

$$\frac{x}{x+8} = e^9$$

Solving for x

To solve for $x$, we can cross-multiply and simplify the equation:

$$x = e^9(x+8)$$

$$x = e^9x + 8e^9$$

Subtracting $e^9x$ from both sides, we get:

$$-e^9x = 8e^9$$

Dividing both sides by $-e^9$, we get:

$$x = -8e^9$$

Approximating the Result to Three Decimal Places

To approximate the result to three decimal places, we can use a calculator to evaluate the expression $-8e^9$.

$$x \approx -8(259.098)$$

$$x \approx -2076.784$$

Conclusion

In this article, we solved the logarithmic equation $\ln x - \ln (x+8) = 9$ algebraically and approximated the result to three decimal places. We used the properties of logarithms and exponential functions to eliminate the logarithm and solve for $x$. The final result is $x \approx -2076.784$.

Final Answer

The final answer is $\boxed{-2076.784}$.

Additional Resources

For more information on logarithmic equations and exponential functions, please refer to the following resources:

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

Discussion

Introduction

In our previous article, we solved the logarithmic equation $\ln x - \ln (x+8) = 9$ algebraically and approximated the result to three decimal places. In this article, we will provide a Q&A guide to help you better understand the concept of solving logarithmic equations algebraically.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The logarithmic function is defined as $\log_b a = c$, where $b$ is the base of the logarithm, $a$ is the argument of the logarithm, and $c$ is the result of the logarithm.

Q: How do I solve a logarithmic equation algebraically?

A: To solve a logarithmic equation algebraically, you can use the properties of logarithms, such as the property that $\ln a - \ln b = \ln \frac{a}{b}$. You can also use the exponential function to eliminate the logarithm.

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

A: A logarithmic equation is an equation that involves a logarithmic function, while an exponential equation is an equation that involves an exponential function. For example, the equation $\ln x = 9$ is a logarithmic equation, while the equation $e^x = 9$ is an exponential equation.

Q: How do I use the properties of logarithms to solve a logarithmic equation?

A: To use the properties of logarithms to solve a logarithmic equation, you can use the following properties:

• $\ln a - \ln b = \ln \frac{a}{b}$
• $\ln a + \ln b = \ln (ab)$
• $\ln a^b = b\ln a$

You can use these properties to simplify the equation and solve for the variable.

Q: What is the base of the logarithm?

A: The base of the logarithm is the number that is used to define the logarithmic function. For example, the base of the natural logarithm is $e$, while the base of the common logarithm is $10$.

Q: How do I approximate the result of a logarithmic equation to three decimal places?

A: To approximate the result of a logarithmic equation to three decimal places, you can use a calculator to evaluate the expression. You can also use the properties of logarithms to simplify the equation and approximate the result.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not using the properties of logarithms to simplify the equation
• Not eliminating the logarithm using the exponential function
• Not approximating the result to three decimal places

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of solving logarithmic equations algebraically. We covered topics such as the definition of a logarithmic equation, the properties of logarithms, and common mistakes to avoid. We hope this guide has been helpful in your understanding of logarithmic equations.

Final Answer

The final answer is $\boxed{NO SOLUTION}$.

Additional Resources

For more information on logarithmic equations and exponential functions, please refer to the following resources:

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

Discussion

Do you have any questions or comments about solving logarithmic equations algebraically? Please share your thoughts in the discussion section below.