Find The Solution To The Equation Log₂x + Log₂(x-6) = 4.

by ADMIN 57 views

This article delves into finding the solution to the logarithmic equation log₂x + log₂(x-6) = 4. Logarithmic equations can seem daunting at first, but by understanding the fundamental properties of logarithms and applying algebraic techniques, we can systematically solve them. In this comprehensive guide, we will walk through each step, ensuring a clear understanding of the process. Mastering logarithmic equations is crucial not only for academic pursuits in mathematics but also for various applications in science, engineering, and finance. By the end of this article, you will have a solid grasp of how to solve such equations and be able to tackle similar problems with confidence. Logarithmic equations, deeply rooted in the principles of exponents and inverse functions, play a pivotal role in various scientific and mathematical contexts. They are instrumental in modeling natural phenomena, such as exponential growth and decay, which appear in fields like biology, physics, and economics. The equation at hand, log₂x + log₂(x - 6) = 4, provides a perfect platform to demonstrate the techniques involved in solving logarithmic equations, offering a pathway for readers to deepen their understanding and enhance their mathematical abilities. This exploration will cover logarithmic properties, algebraic manipulation, and critical analysis to ensure the solutions are valid within the domain of the logarithmic functions. The solution to this equation unveils the value of x that satisfies the given relationship between logarithmic functions. Throughout the following sections, we will meticulously break down each step, offering clear explanations and insights to illuminate the process of solving logarithmic equations effectively. The journey through this equation promises not just an answer but a profound understanding of logarithmic functions and their applications. In the subsequent sections, we will embark on a detailed exploration of the mathematical techniques required to solve this equation, ensuring that each step is carefully explained and justified. This approach will enable readers to not only find the solution but also to comprehend the underlying principles, fostering a deeper appreciation for mathematics and its applications. By meticulously applying the properties of logarithms and engaging in rigorous algebraic manipulation, we will unravel the value of x that harmonizes with the given equation, thereby providing a comprehensive understanding of the intricacies involved in solving logarithmic equations.

Step-by-Step Solution

1. Combine Logarithms

Our initial step in solving the equation log₂x + log₂(x-6) = 4 involves combining the logarithmic terms on the left side. We can utilize the logarithmic property that states: logₐ(m) + logₐ(n) = logₐ(mn). This property allows us to condense two logarithmic terms with the same base into a single logarithm. Applying this rule to our equation, we get:

log₂(x(x-6)) = 4

This simplification is crucial as it transforms the equation into a more manageable form. By consolidating the two logarithms into one, we effectively reduce the complexity of the problem and set the stage for further algebraic manipulations. This initial transformation underscores the significance of logarithmic properties in streamlining the equation-solving process. The combined logarithmic term, log₂(x(x - 6)), now represents the product of x and (x - 6) within the logarithm, which will be crucial for our next steps. This consolidation enables us to handle the equation more efficiently, paving the way for converting the logarithmic form into an exponential form. Mastering this step is essential for anyone looking to solve logarithmic equations with proficiency, as it simplifies the overall process and sets the groundwork for further transformations and solutions. Understanding how to combine logarithms is a fundamental skill in handling such equations, and it allows for a more intuitive approach to solving complex mathematical problems involving logarithmic functions. The combined form, log₂(x(x - 6)) = 4, is not only easier to work with but also provides a clearer path toward isolating the variable x, which is our ultimate goal. This step exemplifies the power of logarithmic properties in transforming complex expressions into simpler, solvable forms, highlighting their importance in mathematical problem-solving.

2. Convert to Exponential Form

Having combined the logarithmic terms, our next move is to convert the equation from logarithmic form to exponential form. This conversion is a pivotal step in solving logarithmic equations, allowing us to eliminate the logarithm and express the relationship in a more familiar algebraic format. The fundamental relationship between logarithms and exponentials states that if logₐ(b) = c, then aᶜ = b. Applying this principle to our equation, log₂(x(x-6)) = 4, we can rewrite it as:

2⁴ = x(x-6)

This transformation is significant because it frees the variable x from the confines of the logarithm, making it possible to solve for x directly. The exponential form, 2⁴ = x(x - 6), is much easier to manipulate algebraically compared to the original logarithmic form. This conversion step highlights the inverse relationship between logarithmic and exponential functions, a cornerstone concept in mathematics. By understanding and applying this relationship, we can simplify complex equations and make them solvable using standard algebraic techniques. The process of converting to exponential form not only aids in solving equations but also reinforces the connection between logarithms and exponents, fostering a deeper understanding of mathematical functions. This conversion bridges the gap between logarithmic and algebraic expressions, allowing us to apply a wide range of algebraic tools to find the solution. The equation now in exponential form, 2⁴ = x(x - 6), sets the stage for simplifying and solving for x, moving us closer to our goal of finding the solution to the original logarithmic equation. This step underscores the importance of understanding the fundamental definitions and properties of logarithms and exponents in solving mathematical problems effectively.

3. Simplify and Rearrange

Following the conversion to exponential form, our next step involves simplifying the equation and rearranging it into a standard quadratic form. First, let's evaluate 2⁴, which is 16. Our equation now reads:

16 = x(x-6)

Next, we need to expand the right side of the equation by distributing x across the terms inside the parentheses:

16 = x² - 6x

Now, to solve for x, we need to rearrange the equation into the standard quadratic form, which is ax² + bx + c = 0. To achieve this, we subtract 16 from both sides of the equation:

x² - 6x - 16 = 0

This rearrangement is crucial because it allows us to apply standard methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Transforming the equation into standard quadratic form is a fundamental technique in algebra, providing a structured approach to finding solutions. The simplified and rearranged equation, x² - 6x - 16 = 0, is now in a format that is amenable to various solution methods, making it easier to find the values of x that satisfy the original logarithmic equation. This step demonstrates the power of algebraic manipulation in transforming equations into more solvable forms. By simplifying and rearranging, we have effectively prepared the equation for the next phase of the solution process, which involves finding the roots of the quadratic equation. The transition to the standard quadratic form underscores the importance of algebraic skills in solving mathematical problems, highlighting how rearranging terms can unlock solutions that may not be immediately apparent. This process not only helps in solving the current equation but also reinforces fundamental algebraic principles that are applicable across a wide range of mathematical contexts.

4. Solve the Quadratic Equation

Now that we have the quadratic equation in standard form, x² - 6x - 16 = 0, we can proceed to solve for x. One common method for solving quadratic equations is factoring. We look for two numbers that multiply to -16 and add up to -6. These numbers are -8 and 2. Therefore, we can factor the quadratic equation as follows:

(x - 8)(x + 2) = 0

To find the solutions, we set each factor equal to zero:

x - 8 = 0 or x + 2 = 0

Solving these linear equations, we get:

x = 8 or x = -2

Thus, we have two potential solutions for x: 8 and -2. Factoring is a powerful technique for solving quadratic equations, especially when the roots are integers. It simplifies the process of finding the solutions by breaking the quadratic expression into two linear factors. The factored form, (x - 8)(x + 2) = 0, makes it easy to identify the roots of the equation, as each factor corresponds to a potential solution. This step demonstrates the elegance of factoring in simplifying algebraic expressions and solving equations. By finding the roots of the quadratic equation, we have identified the values of x that could potentially satisfy the original logarithmic equation. However, it is crucial to remember that we must verify these solutions to ensure they are valid within the domain of the logarithmic functions. The solutions x = 8 and x = -2 are candidates, but they need to be checked against the original equation to avoid extraneous solutions. This step in the solution process highlights the importance of not only solving equations but also verifying the solutions to ensure their validity. The factoring method showcases the beauty of algebraic manipulation and its effectiveness in simplifying complex expressions and finding solutions to quadratic equations.

5. Check for Extraneous Solutions

An absolutely crucial step in solving logarithmic equations is checking for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but do not satisfy the original equation. This often occurs because the domain of logarithmic functions is restricted to positive arguments. Our original equation was:

log₂x + log₂(x-6) = 4

We found two potential solutions: x = 8 and x = -2. Let's check each one:

  • For x = 8:

    log₂(8) + log₂(8-6) = log₂(8) + log₂(2) = 3 + 1 = 4

    This solution works.

  • For x = -2:

    log₂(-2) + log₂(-2-6) = log₂(-2) + log₂(-8)

    Logarithms of negative numbers are not defined in the real number system. Therefore, x = -2 is an extraneous solution.

Checking for extraneous solutions is essential because logarithmic functions are only defined for positive arguments. Substituting a negative value or a value that results in a negative argument inside the logarithm will lead to an undefined expression. This step underscores the importance of understanding the domain of logarithmic functions and verifying solutions to ensure they are valid. The extraneous solution check highlights the necessity of returning to the original equation to confirm that the solutions obtained are legitimate. By verifying our solutions, we avoid making errors and ensure the accuracy of our result. The process of checking for extraneous solutions reinforces the concept of domain restrictions in mathematics and the importance of careful verification in problem-solving. This step is not just a formality but a critical component of solving logarithmic equations, ensuring that we arrive at the correct solution. The identification of x = -2 as an extraneous solution demonstrates the practical significance of this verification step, emphasizing its role in preventing incorrect conclusions.

Final Answer

After carefully checking for extraneous solutions, we find that the only valid solution to the equation log₂x + log₂(x-6) = 4 is x = 8.

In this detailed exploration, we successfully navigated the process of solving the logarithmic equation log₂x + log₂(x-6) = 4. We began by leveraging the properties of logarithms to combine logarithmic terms, simplifying the equation into a more manageable form. Subsequently, we converted the logarithmic equation into an exponential equation, which allowed us to free the variable x from the logarithmic constraints. This critical step paved the way for algebraic manipulation, where we rearranged the equation into a standard quadratic form. Solving the resulting quadratic equation yielded two potential solutions, but the importance of checking for extraneous solutions became apparent as one solution was found to be invalid due to the domain restrictions of logarithmic functions. The methodical approach we employed underscores the significance of understanding the fundamental principles of logarithms and algebra, as well as the necessity of meticulous verification to ensure the accuracy of our solutions. By walking through each step with clarity and precision, we have not only found the solution to the given equation but also reinforced valuable problem-solving skills applicable to a wide range of mathematical challenges. This exploration serves as a testament to the power of systematic problem-solving and the importance of adhering to the rules and properties that govern mathematical functions. The process of solving logarithmic equations, as demonstrated in this article, is a blend of algebraic techniques and a deep understanding of the characteristics of logarithmic functions. From combining logarithmic terms to converting to exponential form, each step requires careful application of mathematical principles. The final verification step, checking for extraneous solutions, is crucial in ensuring the correctness of the result and highlights the importance of considering domain restrictions. This comprehensive journey through the solution of log₂x + log₂(x-6) = 4 not only provides the answer but also imparts valuable insights into mathematical problem-solving, reinforcing the idea that a combination of knowledge, technique, and precision is the key to success in mathematics. The final solution, x = 8, stands as a testament to the efficacy of the methods employed and the importance of a thorough and systematic approach to solving mathematical problems. Understanding and applying these principles will undoubtedly empower readers to confidently tackle future challenges in mathematics and beyond.