Solving The Logarithmic Equation Ln(x) + Ln(x-3) = Ln(6x) A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of solving the logarithmic equation ln(x) + ln(x-3) = ln(6x). Logarithmic equations are a fundamental topic in mathematics, frequently encountered in algebra and calculus. Understanding how to solve these equations is crucial for various applications in science, engineering, and finance. This article provides a step-by-step guide, explaining the underlying principles and potential pitfalls, ensuring a comprehensive understanding of the solution.

Solving logarithmic equations involves several key steps. First, we need to apply the properties of logarithms to simplify the equation. Specifically, we'll use the property that the sum of logarithms is the logarithm of the product. Then, we will convert the logarithmic equation into an algebraic equation, which can be solved using standard algebraic techniques. Finally, and most importantly, we must verify the solutions obtained to ensure they are valid within the domain of the original logarithmic equation. Logarithmic functions are only defined for positive arguments, so any solution that results in a negative or zero argument must be discarded. Let's embark on this detailed journey to master the art of solving logarithmic equations.

Step-by-Step Solution

1. Combine Logarithms

The initial step in solving the equation ln(x) + ln(x-3) = ln(6x) is to combine the logarithms on the left-hand side. We will use the logarithmic property that states the sum of logarithms is equal to the logarithm of the product. This property is expressed as: ln(a) + ln(b) = ln(ab). Applying this property to our equation, we combine the two logarithmic terms on the left side into a single logarithm:

ln(x) + ln(x - 3) = ln(x(x - 3))

This simplifies the left-hand side of our equation, making it easier to work with. Now, our equation looks like this:

ln(x(x - 3)) = ln(6x)

By combining the logarithms, we've reduced the complexity of the equation and set the stage for the next step: eliminating the logarithms.

2. Eliminate Logarithms

Now that we have a single logarithm on each side of the equation ln(x(x - 3)) = ln(6x), we can eliminate the logarithms. This is achieved by using the property that if ln(a) = ln(b), then a = b. In other words, if the natural logarithms of two expressions are equal, then the expressions themselves must be equal. Applying this property to our equation, we get:

x(x - 3) = 6x

This step transforms our logarithmic equation into a standard algebraic equation, specifically a quadratic equation. This transition is crucial because we can now leverage our knowledge of algebraic techniques to solve for x. Eliminating the logarithms allows us to work with a more familiar form of the equation, making it easier to find potential solutions.

3. Solve the Quadratic Equation

After eliminating the logarithms, we now have the equation x(x - 3) = 6x. The next step is to solve this quadratic equation for x. First, we expand the left-hand side:

x^2 - 3x = 6x

Next, we want to set the equation equal to zero, which is the standard form for solving quadratic equations. Subtract 6x from both sides of the equation:

x^2 - 3x - 6x = 0

x^2 - 9x = 0

Now we can factor out an x from the left-hand side:

x(x - 9) = 0

This gives us two possible solutions for x: either x = 0 or x - 9 = 0, which means x = 9. So, our potential solutions are:

  • x = 0
  • x = 9

However, it's crucial to remember that these are just potential solutions. We must verify them against the original logarithmic equation to ensure they are valid.

4. Verify Solutions

Verifying the solutions is a critical step in solving logarithmic equations. Since logarithmic functions are only defined for positive arguments, we must check if our potential solutions, x = 0 and x = 9, result in taking the logarithm of a non-positive number in the original equation ln(x) + ln(x - 3) = ln(6x).

Let's start by checking x = 0:

If we substitute x = 0 into the original equation, we get ln(0) + ln(0 - 3) = ln(6 * 0). Since the logarithm of 0 is undefined, and ln(-3) is also undefined, x = 0 is not a valid solution. It falls outside the domain of the logarithmic functions in the equation.

Now, let's check x = 9:

Substitute x = 9 into the original equation:

ln(9) + ln(9 - 3) = ln(6 * 9)

ln(9) + ln(6) = ln(54)

Using the logarithmic property ln(a) + ln(b) = ln(ab), we can rewrite the left side:

ln(9 * 6) = ln(54)

ln(54) = ln(54)

This equality holds true, so x = 9 is a valid solution. It satisfies the original logarithmic equation without resulting in the logarithm of a non-positive number.

Conclusion

In conclusion, the solution to the logarithmic equation ln(x) + ln(x - 3) = ln(6x) is x = 9. We arrived at this solution by systematically applying the properties of logarithms, converting the equation into a quadratic equation, and, most importantly, verifying the solutions. The process involved:

  1. Combining logarithms using the property ln(a) + ln(b) = ln(ab).
  2. Eliminating logarithms by equating the arguments when the logarithms are equal.
  3. Solving the resulting quadratic equation using factoring techniques.
  4. Verifying the solutions to ensure they are within the domain of the original logarithmic equation.

The critical step of verifying the solutions highlights the importance of understanding the domain restrictions of logarithmic functions. Discarding extraneous solutions is crucial for obtaining the correct answer. This comprehensive approach ensures an accurate and thorough solution to logarithmic equations.

Additional Tips for Solving Logarithmic Equations

  • Always Check for Extraneous Solutions: As demonstrated in the solution above, it is essential to verify your solutions in the original equation. Logarithmic functions have domain restrictions (arguments must be positive), so some solutions obtained algebraically may not be valid.
  • Use Logarithmic Properties: Familiarize yourself with the properties of logarithms, such as the product rule (ln(ab) = ln(a) + ln(b)), quotient rule (ln(a/b) = ln(a) - ln(b)), and power rule (ln(a^n) = n ln(a)). These properties are crucial for simplifying and solving logarithmic equations.
  • Convert to Exponential Form: If you have an equation of the form ln(x) = a, you can convert it to exponential form e^a = x to solve for x. Similarly, for other bases, if log_b(x) = a, then b^a = x.
  • Isolate Logarithmic Terms: Before applying any properties or converting to exponential form, isolate the logarithmic terms on one side of the equation. This often involves algebraic manipulation to group the logarithmic terms together.
  • Be Mindful of the Domain: Always consider the domain of the logarithmic functions in the equation. The arguments of the logarithms must be positive. This will help you identify potential extraneous solutions early in the process.
  • Use Substitution: For more complex equations, you might find it helpful to use substitution. For example, if you have an equation with ln(x) and (ln(x))^2, you could substitute y = ln(x) to create a quadratic equation in y.
  • Graphical Solutions: If you're having trouble solving algebraically, you can use a graphing calculator or software to graph the functions on both sides of the equation and find the points of intersection. This can provide a visual confirmation of your algebraic solutions.
  • Practice Regularly: The best way to improve your skills in solving logarithmic equations is to practice regularly. Work through a variety of problems, and pay attention to the different techniques and strategies that can be applied.

By following these tips and practicing consistently, you can enhance your ability to solve logarithmic equations effectively and accurately.

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to make mistakes if you're not careful. Recognizing these common pitfalls can help you avoid them and improve your accuracy. Here are some frequent errors to watch out for:

  • Forgetting to Check for Extraneous Solutions: This is arguably the most common mistake. As we've emphasized, solutions obtained algebraically must be verified in the original equation due to the domain restrictions of logarithmic functions. Failing to do so can lead to incorrect answers.
  • Incorrectly Applying Logarithmic Properties: Logarithmic properties are powerful tools, but they must be applied correctly. For example, ln(a + b) is not equal to ln(a) + ln(b). Make sure you understand and apply the properties accurately.
  • Ignoring the Domain: The domain of a logarithmic function is crucial. Remember that the argument of a logarithm must be positive. Ignoring this can lead to including solutions that are not valid.
  • Dividing by a Variable Expression: When solving equations, avoid dividing both sides by an expression that contains a variable (e.g., x). This can lead to losing solutions. Instead, move all terms to one side and factor.
  • Misunderstanding the Change of Base Formula: If you need to change the base of a logarithm, make sure you apply the change of base formula correctly. The formula is: log_b(a) = ln(a) / ln(b).
  • Incorrectly Converting to Exponential Form: When converting a logarithmic equation to exponential form, ensure you understand the relationship between the base, exponent, and argument. For example, if log_b(a) = c, then b^c = a.
  • Making Algebraic Errors: Solving logarithmic equations often involves algebraic manipulation, such as expanding, factoring, and simplifying. Be careful with your algebra to avoid errors that can lead to incorrect solutions.
  • Skipping Steps: It's tempting to skip steps to save time, but this can increase the likelihood of making mistakes. Write out each step clearly to minimize errors.

By being aware of these common mistakes and taking precautions to avoid them, you can improve your accuracy and confidence in solving logarithmic equations. Always double-check your work and verify your solutions to ensure they are correct.