Solve For \[$ X \$\].$\[ \ln (4x - 5) - \ln (8x) = \ln 2 \\]

Introduction

In this article, we will delve into solving a logarithmic equation involving natural logarithms. The given equation is $\ln (4x - 5) - \ln (8x) = \ln 2$. Our goal is to isolate the variable $x$ and find its value. We will use properties of logarithms and algebraic manipulations to solve for $x$.

Understanding the Equation

The given equation involves natural logarithms, which are denoted by $\ln$. The equation is $\ln (4x - 5) - \ln (8x) = \ln 2$. To begin solving this equation, we need to understand the properties of logarithms. One of the key properties is the quotient rule, which states that $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$.

Applying the Quotient Rule

Using the quotient rule, we can rewrite the given equation as $\ln \left(\frac{4x - 5}{8x}\right) = \ln 2$. This simplifies the equation and makes it easier to work with.

Exponentiating Both Sides

To get rid of the logarithm, we can exponentiate both sides of the equation. Since the base of the natural logarithm is $e$, we can raise $e$ to both sides of the equation. This gives us $e^{\ln \left(\frac{4x - 5}{8x}\right)} = e^{\ln 2}$.

Simplifying the Exponents

Using the property of exponents that $e^{\ln a} = a$, we can simplify the exponents on both sides of the equation. This gives us $\frac{4x - 5}{8x} = 2$.

Solving for $x$

Now that we have a simplified equation, we can solve for $x$. To do this, we can start by multiplying both sides of the equation by $8x$ to eliminate the fraction. This gives us $4x - 5 = 16x$.

Isolating $x$

Next, we can isolate $x$ by moving all the terms involving $x$ to one side of the equation. We can do this by subtracting $4x$ from both sides of the equation. This gives us $-5 = 12x$.

Final Step

Finally, we can solve for $x$ by dividing both sides of the equation by $12$. This gives us $x = -\frac{5}{12}$.

Conclusion

In this article, we solved a logarithmic equation involving natural logarithms. We used properties of logarithms and algebraic manipulations to isolate the variable $x$ and find its value. The final solution is $x = -\frac{5}{12}$.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. $\ln (4x - 5) - \ln (8x) = \ln 2$
2. $\ln \left(\frac{4x - 5}{8x}\right) = \ln 2$
3. $e^{\ln \left(\frac{4x - 5}{8x}\right)} = e^{\ln 2}$
4. $\frac{4x - 5}{8x} = 2$
5. $4x - 5 = 16x$
6. $-5 = 12x$
7. $x = -\frac{5}{12}$

Final Answer

The final answer is $x = -\frac{5}{12}$.

Introduction

In our previous article, we solved a logarithmic equation involving natural logarithms. The given equation was $\ln (4x - 5) - \ln (8x) = \ln 2$. We used properties of logarithms and algebraic manipulations to isolate the variable $x$ and find its value. In this article, we will answer some frequently asked questions related to the solution of the equation.

Q&A

Q: What is the main property of logarithms used in solving the equation?

A: The main property of logarithms used in solving the equation is the quotient rule, which states that $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$.

Q: How do we get rid of the logarithm in the equation?

A: We can get rid of the logarithm in the equation by exponentiating both sides of the equation. Since the base of the natural logarithm is $e$, we can raise $e$ to both sides of the equation.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = -\frac{5}{12}$.

Q: Can you explain the step-by-step solution to the equation?

A: Here is the step-by-step solution to the equation:

1. $\ln (4x - 5) - \ln (8x) = \ln 2$
2. $\ln \left(\frac{4x - 5}{8x}\right) = \ln 2$
3. $e^{\ln \left(\frac{4x - 5}{8x}\right)} = e^{\ln 2}$
4. $\frac{4x - 5}{8x} = 2$
5. $4x - 5 = 16x$
6. $-5 = 12x$
7. $x = -\frac{5}{12}$

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 correct properties of logarithms
• Not simplifying the equation properly
• Not isolating the variable correctly
• Not checking the solution for extraneous solutions

Q: Can you provide some examples of logarithmic equations that can be solved using the same method?

A: Yes, here are some examples of logarithmic equations that can be solved using the same method:

• $\ln (3x - 2) - \ln (6x) = \ln 4$
• $\ln (2x + 1) - \ln (4x) = \ln 3$
• $\ln (x - 1) - \ln (2x) = \ln 2$

Conclusion

In this article, we answered some frequently asked questions related to the solution of the logarithmic equation $\ln (4x - 5) - \ln (8x) = \ln 2$. We provided step-by-step solutions to the equation and discussed some common mistakes to avoid when solving logarithmic equations. We also provided some examples of logarithmic equations that can be solved using the same method.

Final Answer

The final answer is $x = -\frac{5}{12}$.