Solve The Equation: $\ln 2 - \ln (3x + 2) = 1$

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Introduction

In this article, we will delve into solving a logarithmic equation involving natural logarithms. The equation in question is $\ln 2 - \ln (3x + 2) = 1$. We will use properties of logarithms to simplify the equation and solve for the variable $x$. This equation is a great example of how logarithmic properties can be used to solve complex equations.

Understanding Logarithmic Properties

Before we dive into solving the equation, let's review some key logarithmic properties that we will use:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

These properties will be essential in simplifying the equation and solving for $x$.

Simplifying the Equation

Let's start by simplifying the equation using the quotient property of logarithms:

$$\ln 2 - \ln (3x + 2) = 1$$

Using the quotient property, we can rewrite the equation as:

$$\ln \frac{2}{3x + 2} = 1$$

Now, let's use the definition of a logarithm to rewrite the equation in exponential form:

$$\frac{2}{3x + 2} = e^1$$

Since $e^1 = e$, we can rewrite the equation as:

$$\frac{2}{3x + 2} = e$$

Solving for $x$

Now that we have the equation in exponential form, we can solve for $x$. Let's start by isolating the term with $x$:

$$3x + 2 = \frac{2}{e}$$

Subtracting 2 from both sides gives us:

$$3x = \frac{2}{e} - 2$$

Dividing both sides by 3 gives us:

$$x = \frac{\frac{2}{e} - 2}{3}$$

Simplifying the Expression

Let's simplify the expression for $x$:

$$x = \frac{\frac{2}{e} - 2}{3}$$

Using the fact that $e \approx 2.718$, we can rewrite the expression as:

$$x = \frac{\frac{2}{2.718} - 2}{3}$$

Simplifying the expression gives us:

$$x = \frac{0.735 - 2}{3}$$

$$x = \frac{-1.265}{3}$$

$$x \approx -0.422$$

Conclusion

In this article, we solved the equation $\ln 2 - \ln (3x + 2) = 1$ using properties of logarithms. We simplified the equation using the quotient property and then solved for $x$ using the definition of a logarithm. The final solution is $x \approx -0.422$. This equation is a great example of how logarithmic properties can be used to solve complex equations.

Additional Resources

For more information on logarithmic properties and how to solve logarithmic equations, check out the following resources:

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

Frequently Asked Questions

Q: What is the definition of a logarithm? A: The definition of a logarithm is the inverse operation of exponentiation. In other words, if $y = b^x$, then $x = \log_b y$.

Q: What is the quotient property of logarithms? A: The quotient property of logarithms states that $\log_b \frac{x}{y} = \log_b x - \log_b y$.

Introduction

Logarithmic equations can be challenging to solve, but with the right tools and techniques, you can master them. In this article, we will provide a comprehensive Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. It is an equation that can be written in the form $\log_b x = y$ or $\log_b x = \log_b y$.

Q: What are the different types of logarithmic equations?

A: There are two main types of logarithmic equations:

• Logarithmic equations with a single logarithm: These equations involve a single logarithm, such as $\log_b x = y$.
• Logarithmic equations with multiple logarithms: These equations involve multiple logarithms, such as $\log_b x + \log_b y = z$.

Q: How do I solve a logarithmic equation with a single logarithm?

A: To solve a logarithmic equation with a single logarithm, you can use the definition of a logarithm to rewrite the equation in exponential form. For example, if you have the equation $\log_b x = y$, you can rewrite it as $b^y = x$.

Q: How do I solve a logarithmic equation with multiple logarithms?

A: To solve a logarithmic equation with multiple logarithms, you can use the product property and quotient property of logarithms to simplify the equation. For example, if you have the equation $\log_b x + \log_b y = z$, you can rewrite it as $\log_b (xy) = z$.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_b (xy) = \log_b x + \log_b y$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_b \frac{x}{y} = \log_b x - \log_b y$.

Q: How do I use the product property and quotient property to simplify a logarithmic equation?

A: To use the product property and quotient property to simplify a logarithmic equation, you can follow these steps:

1. Identify the logarithmic terms in the equation.
2. Use the product property to combine the logarithmic terms.
3. Use the quotient property to simplify the equation.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation $\log_b x = y$ is a logarithmic equation, while the equation $b^x = y$ is an exponential equation.

Q: How do I convert a logarithmic equation to an exponential equation?

A: To convert a logarithmic equation to an exponential equation, you can use the definition of a logarithm to rewrite the equation in exponential form. For example, if you have the equation $\log_b x = y$, you can rewrite it as $b^y = x$.

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

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

• Forgetting to use the product property and quotient property to simplify the equation.
• Not using the definition of a logarithm to rewrite the equation in exponential form.
• Not checking the domain of the logarithmic function.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand logarithmic equations and how to solve them. We have covered topics such as the definition of a logarithmic equation, the product property and quotient property of logarithms, and how to convert a logarithmic equation to an exponential equation. By following the tips and techniques outlined in this article, you can master logarithmic equations and become a proficient problem-solver.

Additional Resources

For more information on logarithmic equations and how to solve them, check out the following resources:

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

Frequently Asked Questions

Q: What is the definition of a logarithmic equation? A: A logarithmic equation is an equation that involves a logarithm.

Q: What are the different types of logarithmic equations? A: There are two main types of logarithmic equations: logarithmic equations with a single logarithm and logarithmic equations with multiple logarithms.

Q: How do I solve a logarithmic equation with a single logarithm? A: To solve a logarithmic equation with a single logarithm, you can use the definition of a logarithm to rewrite the equation in exponential form.

Q: How do I solve a logarithmic equation with multiple logarithms? A: To solve a logarithmic equation with multiple logarithms, you can use the product property and quotient property of logarithms to simplify the equation.