Extraneous Solutions In Logarithmic Equations Solving Log₄(x) + Log₄(x-3) = Log₄(-7x+21)
In the realm of mathematics, particularly when dealing with logarithmic equations, the concept of extraneous solutions often presents a fascinating challenge. These are solutions that emerge during the solving process but do not satisfy the original equation. This article delves into the extraneous solution of the logarithmic equation log₄(x) + log₄(x-3) = log₄(-7x+21), a common topic in algebra and precalculus courses. We will methodically dissect the equation, solve for potential solutions, and then rigorously test them to identify any extraneous solutions. Understanding extraneous solutions is crucial for students and enthusiasts alike as it reinforces the importance of validating solutions within the original context of the problem.
Understanding Logarithmic Equations
Before diving into the specific equation, it’s essential to grasp the fundamentals of logarithmic equations. A logarithm is essentially the inverse operation to exponentiation. The logarithmic expression logₐ(b) = c can be rewritten in exponential form as aᶜ = b. Here, 'a' is the base of the logarithm, 'b' is the argument, and 'c' is the exponent. A critical condition for logarithms is that the argument must be strictly positive. This condition arises from the definition of logarithms in relation to exponential functions, which are only defined for positive arguments. Thus, for logₐ(b) to be valid, 'b' must be greater than zero.
This positivity requirement is where extraneous solutions often sneak into the picture. When solving logarithmic equations, we frequently use properties of logarithms to combine or simplify expressions. These manipulations can sometimes lead to solutions that, when plugged back into the original equation, result in taking the logarithm of a negative number or zero. Such solutions are deemed extraneous because they violate the fundamental condition of logarithmic functions. Recognizing and identifying these extraneous solutions is a vital skill in solving logarithmic equations correctly.
Furthermore, understanding the domain of logarithmic functions is paramount. The domain of a logarithmic function logₐ(x) is all positive real numbers (x > 0). When dealing with more complex logarithmic equations, such as the one we are examining, this domain restriction applies to every logarithmic term in the equation. Therefore, each argument within a logarithmic function must be greater than zero. This principle guides us in setting up inequalities to determine the valid range of solutions even before we begin the solving process. By keeping this domain restriction in mind, we can proactively identify potential extraneous solutions and avoid unnecessary calculations.
Solving the Equation log₄(x) + log₄(x-3) = log₄(-7x+21)
To solve the logarithmic equation log₄(x) + log₄(x-3) = log₄(-7x+21), we will follow a step-by-step approach, employing the properties of logarithms and algebraic techniques. First, we need to recognize the fundamental property of logarithms that allows us to combine logarithms with the same base. Specifically, logₐ(m) + logₐ(n) = logₐ(mn). Applying this property to the left side of the equation, we can combine the two logarithmic terms into a single logarithm.
Combining the Logarithms: Using the property logₐ(m) + logₐ(n) = logₐ(mn), we can rewrite the left side of the equation as log₄(x(x-3)). Thus, the equation becomes log₄(x(x-3)) = log₄(-7x+21). This step simplifies the equation, making it easier to work with and solve.
Equating the Arguments: Now that we have a single logarithmic term on each side of the equation with the same base, we can equate the arguments. This means that if log₄(x(x-3)) = log₄(-7x+21), then x(x-3) = -7x+21. This step is valid because the logarithmic function is one-to-one, meaning that if the logarithms are equal, their arguments must also be equal. By equating the arguments, we transform the logarithmic equation into a standard algebraic equation, which we can then solve using familiar methods.
Solving the Quadratic Equation: Expanding and simplifying the equation x(x-3) = -7x+21, we get x² - 3x = -7x + 21. Rearranging the terms to form a quadratic equation, we have x² + 4x - 21 = 0. This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We look for two numbers that multiply to -21 and add to 4. These numbers are 7 and -3. Thus, we can factor the quadratic equation as (x+7)(x-3) = 0. Setting each factor equal to zero gives us the potential solutions x = -7 and x = 3.
Identifying Extraneous Solutions
Once we have potential solutions, the next critical step is to check them in the original equation. This is where we determine whether any of the solutions are extraneous. Extraneous solutions are those that satisfy the transformed equation but do not satisfy the original logarithmic equation. To check for extraneous solutions, we substitute each potential solution back into the original equation and verify that the arguments of all logarithms are positive.
Checking x = -7: Substituting x = -7 into the original equation log₄(x) + log₄(x-3) = log₄(-7x+21), we get log₄(-7) + log₄(-7-3) = log₄(-7(-7)+21). This simplifies to log₄(-7) + log₄(-10) = log₄(70). We immediately notice that we are taking the logarithm of negative numbers (-7 and -10), which is undefined. Therefore, x = -7 is an extraneous solution. This highlights the importance of checking solutions in the original equation, especially when dealing with logarithmic functions.
Checking x = 3: Next, we substitute x = 3 into the original equation. This gives us log₄(3) + log₄(3-3) = log₄(-7(3)+21). Simplifying, we have log₄(3) + log₄(0) = log₄(0). Here, we encounter another issue: we are taking the logarithm of zero, which is also undefined. Therefore, x = 3 is also an extraneous solution. This further emphasizes the necessity of verifying solutions to ensure they are valid within the domain of the logarithmic functions involved.
Conclusion: After checking both potential solutions, we find that both x = -7 and x = 3 lead to undefined logarithmic terms in the original equation. Therefore, both solutions are extraneous. This means that the original equation log₄(x) + log₄(x-3) = log₄(-7x+21) has no valid solutions. The process of solving this equation and identifying extraneous solutions underscores the importance of understanding the domain restrictions of logarithmic functions and the necessity of verifying solutions in the original equation.
The Significance of Checking Solutions
The exercise of identifying extraneous solutions in logarithmic equations underscores a fundamental principle in mathematics: the importance of verifying solutions. While algebraic manipulations can lead us to potential answers, these answers are not necessarily valid solutions to the original problem. Checking solutions is particularly crucial when dealing with functions that have domain restrictions, such as logarithmic and radical functions.
In the context of logarithmic equations, the arguments of the logarithms must be positive. This restriction arises from the very definition of logarithms as inverses of exponential functions. Exponential functions always produce positive outputs, so logarithms, which reverse this process, can only accept positive inputs. When we solve logarithmic equations, we often combine or simplify logarithmic terms, which can alter the domain of the equation. This is where extraneous solutions can arise. A solution that satisfies the simplified equation may not satisfy the original equation because it results in taking the logarithm of a negative number or zero.
Moreover, the act of checking solutions reinforces a deeper understanding of the mathematical concepts involved. It compels students and practitioners to engage critically with the problem and to think about the underlying principles that govern the behavior of the functions. This critical engagement is not only valuable for solving problems correctly but also for developing a more robust and intuitive understanding of mathematics.
The implications of extraneous solutions extend beyond the realm of logarithmic equations. They appear in various areas of mathematics, including radical equations, trigonometric equations, and even differential equations. In each of these contexts, the process of solving equations involves transformations that can potentially introduce solutions that do not satisfy the original problem. Therefore, the habit of checking solutions is a universally applicable skill in mathematics.
Conclusion
In conclusion, the logarithmic equation log₄(x) + log₄(x-3) = log₄(-7x+21) provides a compelling example of how extraneous solutions can arise and the critical importance of checking solutions in the original equation. By solving the equation, we found potential solutions of x = -7 and x = 3. However, upon substituting these values back into the original equation, we discovered that both solutions result in taking the logarithm of a non-positive number, making them extraneous.
This exercise highlights several key concepts. First, it reinforces the fundamental requirement that the argument of a logarithm must be positive. Second, it demonstrates how algebraic manipulations, while necessary for solving equations, can sometimes lead to solutions that are not valid in the original context. Finally, it underscores the necessity of verifying solutions to ensure their validity, especially when dealing with functions that have domain restrictions.
Understanding and identifying extraneous solutions is a crucial skill for anyone studying mathematics, particularly in algebra and precalculus. It not only helps in solving problems correctly but also fosters a deeper appreciation for the underlying principles and constraints of mathematical functions. By consistently checking solutions, students can avoid common pitfalls and develop a more rigorous and intuitive understanding of mathematics. This careful approach not only leads to correct answers but also enhances problem-solving skills and mathematical reasoning, preparing students for more advanced topics in mathematics and related fields.
Therefore, the exploration of extraneous solutions in logarithmic equations serves as a valuable lesson in mathematical rigor and the importance of critical thinking. It encourages a proactive approach to problem-solving, where potential solutions are always subjected to scrutiny and verification, ensuring that the final answer is not only mathematically correct but also logically sound within the context of the problem.
