Solve The Equation Log₂x + Log₂(x-6) = 4 For X.

Jun 19, 2025 by ADMIN 48 views

Solving the Logarithmic Equation: log₂x + log₂(x-6) = 4

Introduction: Mastering Logarithmic Equations

In the realm of mathematics, solving equations is a fundamental skill, and this is especially true when dealing with logarithmic equations. Logarithmic equations, characterized by the presence of logarithmic functions, often require a careful application of logarithmic properties and algebraic manipulation to arrive at the correct solution. This article delves into a specific logarithmic equation, guiding you through the step-by-step process of solving it and highlighting key concepts along the way. Our focus is on the equation log₂x + log₂(x-6) = 4, a classic example that demonstrates the importance of understanding logarithmic properties and checking for extraneous solutions. We'll break down each step, providing clear explanations and insights to enhance your problem-solving skills. This comprehensive guide will not only equip you with the tools to tackle similar problems but also deepen your understanding of logarithms and their applications. Whether you're a student preparing for an exam or simply a math enthusiast looking to expand your knowledge, this article offers a valuable resource for mastering logarithmic equations. Understanding the nuances of logarithmic functions is crucial in various fields, including computer science, engineering, and finance, where exponential growth and decay models are prevalent. Therefore, mastering logarithmic equations is not just an academic exercise but a practical skill that can be applied in numerous real-world scenarios.

Problem Statement: Deciphering the Logarithmic Puzzle

The problem at hand is to find the value(s) of x that satisfy the equation:

log₂x + log₂(x-6) = 4

This equation involves logarithms with base 2, and our goal is to isolate x and determine its value(s). To accomplish this, we will leverage the properties of logarithms, particularly the product rule, which allows us to combine the two logarithmic terms on the left-hand side. Additionally, we must consider the domain of logarithmic functions, ensuring that the solutions we obtain are valid. Logarithmic functions are only defined for positive arguments, so we need to check that both x and x-6 are greater than zero. This condition will help us identify any extraneous solutions that may arise during the solving process. The process of solving this equation will demonstrate the power of logarithmic properties in simplifying complex expressions and the importance of careful algebraic manipulation. By working through this problem, you will gain a deeper appreciation for the elegance and versatility of logarithms. Furthermore, you'll develop the critical thinking skills necessary to approach and solve a wide range of logarithmic equations. The problem's structure allows for a clear demonstration of how to combine logarithmic terms, convert logarithmic equations to exponential form, and solve quadratic equations, making it an excellent learning opportunity for anyone studying logarithms.

Solution: A Step-by-Step Journey to the Answer

Apply the Product Rule of Logarithms: The product rule states that logₐ(m) + logₐ(n) = logₐ(mn). Applying this rule to our equation, we get:
```
log₂(x(x-6)) = 4
```
This step is crucial as it combines the two logarithmic terms into a single term, simplifying the equation and making it easier to solve. The product rule is a fundamental property of logarithms, and its application here demonstrates its power in simplifying logarithmic expressions. By combining the terms, we reduce the complexity of the equation and pave the way for the next steps in the solution process. Understanding and applying the product rule is essential for effectively solving logarithmic equations. This step not only simplifies the equation but also sets the stage for converting the logarithmic equation into an exponential equation, which is a key technique in solving logarithmic problems. The careful application of this rule is a testament to the importance of mastering fundamental logarithmic properties.
Convert to Exponential Form: The logarithmic equation log₂[x(x-6)] = 4 can be rewritten in exponential form as:
```
x(x-6) = 2⁴
```
Converting from logarithmic to exponential form is a key step in solving logarithmic equations. It allows us to eliminate the logarithm and work with a more familiar algebraic expression. The base of the logarithm (2) becomes the base of the exponent, and the result of the logarithm (4) becomes the exponent. This transformation is based on the fundamental relationship between logarithms and exponentials: logₐ(b) = c if and only if aᶜ = b. This step simplifies the equation by removing the logarithmic function and transforming it into a polynomial equation, which is easier to solve. Understanding this conversion is crucial for mastering logarithmic equations and is a cornerstone of logarithmic problem-solving techniques. The ability to seamlessly switch between logarithmic and exponential forms is a valuable skill in mathematics and related fields.
Simplify and Rearrange: Calculate 2⁴ and expand the left side of the equation:
```
x(x-6) = 16
x² - 6x = 16
```
Now, move all terms to one side to form a quadratic equation:
```
x² - 6x - 16 = 0
```
This step involves basic algebraic manipulation to transform the equation into a standard quadratic form. Expanding the product on the left side and moving all terms to one side sets up the equation for factoring or using the quadratic formula. The simplified quadratic equation is much easier to solve than the original logarithmic equation, highlighting the effectiveness of the previous steps. The ability to rearrange equations and simplify expressions is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. This step demonstrates the importance of algebraic proficiency in the context of logarithmic equations and underscores the interconnectedness of different mathematical concepts.
Solve the Quadratic Equation: Factor the quadratic equation:
```
(x - 8)(x + 2) = 0
```
This gives us two potential solutions:
```
x = 8  or  x = -2
```
Factoring the quadratic equation is a common technique for finding the roots of a polynomial. The factored form reveals the values of x that make the equation equal to zero, which are the potential solutions to the original equation. Each factor corresponds to a possible solution, and setting each factor equal to zero allows us to find these solutions. The ability to factor quadratic equations is a valuable skill in algebra and is frequently used in various mathematical contexts. This step demonstrates the power of factoring in simplifying equations and finding their solutions, highlighting the importance of mastering factoring techniques.
Check for Extraneous Solutions: Since we are dealing with logarithms, we need to ensure that the arguments of the logarithms are positive. This means we must check if our solutions satisfy the conditions x > 0 and x - 6 > 0.
- For x = 8:
  - log₂(8) is defined (8 > 0)
  - log₂(8-6) = log₂(2) is defined (2 > 0)
  - So, x = 8 is a valid solution.
- For x = -2:
  - log₂(-2) is not defined since the logarithm of a negative number is undefined.
  - Therefore, x = -2 is an extraneous solution.
Checking for extraneous solutions is a crucial step in solving logarithmic equations. Logarithmic functions are only defined for positive arguments, so any solution that results in a non-positive argument is invalid. This step ensures that the solutions we obtain are consistent with the domain of the logarithmic functions involved. Extraneous solutions often arise when squaring both sides of an equation or when dealing with logarithmic or radical equations. Careful checking is necessary to avoid including these invalid solutions in the final answer. This step highlights the importance of understanding the domain of functions and the need for rigorous verification in mathematical problem-solving.

Final Answer: The True Solution Unveiled

Therefore, the only valid solution to the equation log₂x + log₂(x-6) = 4 is:

x = 8

The final answer is the culmination of the step-by-step process we have followed, from applying logarithmic properties to solving a quadratic equation and, most importantly, checking for extraneous solutions. This process underscores the importance of a systematic approach to problem-solving in mathematics. The solution x = 8 satisfies the original equation and the domain restrictions, making it the only valid solution. This final answer represents the successful resolution of the problem and demonstrates the power of combining logarithmic properties with algebraic techniques. The journey to this solution has provided valuable insights into logarithmic equations and their solutions, reinforcing the importance of careful analysis and verification in mathematical problem-solving.

Conclusion: Mastering the Art of Logarithmic Solutions

In conclusion, solving the logarithmic equation log₂x + log₂(x-6) = 4 required a blend of logarithmic properties, algebraic manipulation, and careful consideration of the domain of logarithmic functions. By applying the product rule of logarithms, converting to exponential form, solving the resulting quadratic equation, and checking for extraneous solutions, we arrived at the single valid solution, x = 8. This problem serves as a valuable illustration of the techniques and considerations involved in solving logarithmic equations. Mastering these techniques is essential for anyone working with logarithms in mathematics, science, or engineering. The process highlights the importance of understanding the properties of logarithms and their relationship to exponential functions. Furthermore, it emphasizes the need for careful algebraic manipulation and the critical step of checking for extraneous solutions. The ability to solve logarithmic equations is a fundamental skill that opens doors to a wide range of applications, from modeling exponential growth and decay to analyzing data in scientific research. By practicing and mastering these techniques, you can confidently tackle logarithmic problems and expand your mathematical toolkit. This problem also reinforces the broader mathematical principles of problem-solving, including breaking down complex problems into smaller, manageable steps and the importance of verification in ensuring the accuracy of solutions. The journey from the initial equation to the final answer demonstrates the elegance and power of mathematical reasoning.