Solve The Equation:$\ln (x-3) - \ln (x-5) = \ln 5$

Introduction

In this article, we will delve into the world of mathematics and explore a complex equation involving logarithms. The equation in question is $\ln (x-3) - \ln (x-5) = \ln 5$. Our goal is to solve for the variable $x$ and understand the underlying concepts that make this equation tick.

Understanding Logarithms

Before we dive into the solution, let's take a moment to understand the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $y = a^x$, then $\log_a y = x$. Logarithms have several important properties, including the product rule, which states that $\log_a (xy) = \log_a x + \log_a y$, and the quotient rule, which states that $\log_a \frac{x}{y} = \log_a x - \log_a y$.

Applying the Quotient Rule

The given equation involves the subtraction of two logarithms, which can be rewritten using the quotient rule. Specifically, we have:

$$\ln (x-3) - \ln (x-5) = \ln \frac{x-3}{x-5}$$

This simplifies the equation and allows us to work with a single logarithmic expression.

Equating the Logarithms

Now that we have simplified the equation, we can equate the logarithmic expressions on both sides. This gives us:

$$\ln \frac{x-3}{x-5} = \ln 5$$

Since the logarithmic functions are equal, we can drop the logarithms and equate the arguments:

$$\frac{x-3}{x-5} = 5$$

Solving for x

To solve for $x$, we can start by cross-multiplying:

$$(x-3) = 5(x-5)$$

Expanding the right-hand side, we get:

$$x-3 = 5x - 25$$

Subtracting $x$ from both sides gives us:

$$-3 = 4x - 25$$

Adding 25 to both sides gives us:

$$22 = 4x$$

Dividing both sides by 4 gives us:

$$x = \frac{22}{4}$$

Simplifying the fraction, we get:

$$x = \frac{11}{2}$$

Conclusion

In this article, we have solved the equation $\ln (x-3) - \ln (x-5) = \ln 5$ using the quotient rule and equating the logarithmic expressions. We have also applied algebraic techniques to solve for the variable $x$. The solution is $x = \frac{11}{2}$.

Key Takeaways

• The quotient rule states that $\log_a \frac{x}{y} = \log_a x - \log_a y$.
• The given equation can be rewritten using the quotient rule.
• Equating the logarithmic expressions on both sides allows us to drop the logarithms and equate the arguments.
• Solving for $x$ involves cross-multiplying, expanding, and simplifying the resulting equation.

Real-World Applications

Logarithmic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, logarithmic functions are used to model population growth, chemical reactions, and financial transactions.

Future Directions

In future articles, we can explore more complex logarithmic equations and their applications in various fields. We can also delve into the world of exponential functions and their properties.

References

• [1] "Logarithms" by Khan Academy
• [2] "Exponential Functions" by Mathway
• [3] "Logarithmic Equations" by Wolfram MathWorld

Glossary

• Logarithm: The inverse operation of exponentiation.
• Quotient Rule: A property of logarithms that states $\log_a \frac{x}{y} = \log_a x - \log_a y$.
• Exponential Function: A function of the form $f(x) = a^x$, where $a$ is a positive constant.

Additional Resources

• Khan Academy: Logarithms
• Mathway: Exponential Functions
• Wolfram MathWorld: Logarithmic Equations
  Solving the Equation: A Q&A Guide =====================================

Introduction

In our previous article, we solved the equation $\ln (x-3) - \ln (x-5) = \ln 5$ using the quotient rule and equating the logarithmic expressions. In this article, we will provide a Q&A guide to help you better understand the solution and the underlying concepts.

Q: What is the quotient rule in logarithms?

A: The quotient rule in logarithms states that $\log_a \frac{x}{y} = \log_a x - \log_a y$. This rule allows us to rewrite the subtraction of two logarithms as the logarithm of a quotient.

Q: How do I apply the quotient rule to the given equation?

A: To apply the quotient rule, we can rewrite the subtraction of two logarithms as the logarithm of a quotient. Specifically, we have:

$$\ln (x-3) - \ln (x-5) = \ln \frac{x-3}{x-5}$$

Q: Why can I drop the logarithms and equate the arguments?

A: Since the logarithmic functions are equal, we can drop the logarithms and equate the arguments. This is because the logarithmic functions are one-to-one, meaning that if $\log_a x = \log_a y$, then $x = y$.

Q: How do I solve for x?

A: To solve for $x$, we can start by cross-multiplying:

$$(x-3) = 5(x-5)$$

Expanding the right-hand side, we get:

$$x-3 = 5x - 25$$

Subtracting $x$ from both sides gives us:

$$-3 = 4x - 25$$

Adding 25 to both sides gives us:

$$22 = 4x$$

Dividing both sides by 4 gives us:

$$x = \frac{22}{4}$$

Simplifying the fraction, we get:

$$x = \frac{11}{2}$$

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, logarithmic functions are used to model population growth, chemical reactions, and financial transactions.

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 apply the quotient rule
• Dropping the logarithms without equating the arguments
• Not simplifying the resulting equation
• Not checking the domain of the logarithmic function

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by working through example problems and exercises. You can also try solving logarithmic equations with different bases and arguments.

Q: What are some additional resources for learning about logarithmic equations?

A: Some additional resources for learning about logarithmic equations include:

• Khan Academy: Logarithms
• Mathway: Exponential Functions
• Wolfram MathWorld: Logarithmic Equations

Conclusion

In this article, we have provided a Q&A guide to help you better understand the solution to the equation $\ln (x-3) - \ln (x-5) = \ln 5$. We have also discussed some common mistakes to avoid when solving logarithmic equations and provided some additional resources for learning about logarithmic equations.

Key Takeaways

• The quotient rule states that $\log_a \frac{x}{y} = \log_a x - \log_a y$.
• The given equation can be rewritten using the quotient rule.
• Equating the logarithmic expressions on both sides allows us to drop the logarithms and equate the arguments.
• Solving for $x$ involves cross-multiplying, expanding, and simplifying the resulting equation.

Real-World Applications

Logarithmic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, logarithmic functions are used to model population growth, chemical reactions, and financial transactions.

Future Directions

In future articles, we can explore more complex logarithmic equations and their applications in various fields. We can also delve into the world of exponential functions and their properties.

References

• [1] "Logarithms" by Khan Academy
• [2] "Exponential Functions" by Mathway
• [3] "Logarithmic Equations" by Wolfram MathWorld

Glossary

• Logarithm: The inverse operation of exponentiation.
• Quotient Rule: A property of logarithms that states $\log_a \frac{x}{y} = \log_a x - \log_a y$.
• Exponential Function: A function of the form $f(x) = a^x$, where $a$ is a positive constant.

Additional Resources

• Khan Academy: Logarithms
• Mathway: Exponential Functions
• Wolfram MathWorld: Logarithmic Equations