Extraneous Solution In Logarithmic Equation Log₇(3x³+x) - Log₇(x) = 2

In the realm of mathematics, logarithmic equations present a fascinating challenge, often demanding a meticulous approach to solve them accurately. However, the quest for solutions can sometimes lead us astray, introducing extraneous solutions that appear to satisfy the equation but are, in reality, mathematical illusions. This article delves into the intricacies of extraneous solutions in logarithmic equations, using the equation log₇(3x³+x) - log₇(x) = 2 as a case study. We will explore the concept of extraneous solutions, understand why they arise in logarithmic equations, and develop a systematic approach to identify and eliminate them, ensuring we arrive at the genuine solutions.

Before embarking on the journey of solving the equation log₇(3x³+x) - log₇(x) = 2, it is crucial to grasp the fundamental principles of logarithmic equations and the elusive nature of extraneous solutions. Logarithmic equations involve logarithms, which are the inverse operations of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For instance, log₁₀(100) = 2 because 10² = 100. Logarithmic equations often involve variables within the logarithmic expressions, requiring us to employ logarithmic properties and algebraic manipulations to isolate the variable and find its value.

Extraneous solutions, on the other hand, are solutions that emerge during the solving process but do not satisfy the original equation. They are mathematical mirages, arising from operations that are not reversible or that introduce new possibilities that were not present in the initial equation. In the context of logarithmic equations, extraneous solutions typically arise when we perform operations that alter the domain of the logarithmic functions involved. The domain of a logarithmic function is restricted to positive real numbers, as the logarithm of a non-positive number is undefined. Therefore, any solution that results in a negative or zero argument for any logarithm in the original equation is deemed extraneous.

To illustrate this concept, consider a simplified example: log₁₀(x) + log₁₀(x - 3) = 1. Applying the logarithmic property of addition, we can combine the logarithms: log₁₀[x(x - 3)] = 1. Converting to exponential form, we get x(x - 3) = 10, which simplifies to the quadratic equation x² - 3x - 10 = 0. Factoring the quadratic equation, we obtain (x - 5)(x + 2) = 0, yielding two potential solutions: x = 5 and x = -2. However, if we substitute x = -2 back into the original equation, we encounter log₁₀(-2), which is undefined. Therefore, x = -2 is an extraneous solution, while x = 5 is the only valid solution.

Now, let us embark on the task of solving the given logarithmic equation, log₇(3x³+x) - log₇(x) = 2, while meticulously guarding against the intrusion of extraneous solutions. Our approach will involve employing logarithmic properties to simplify the equation, converting it to an algebraic form, solving the resulting equation, and, most importantly, verifying the solutions to ensure they are not extraneous.

The first step in our solution strategy is to utilize the logarithmic property of subtraction, which states that logₐ(b) - logₐ(c) = logₐ(b/c). Applying this property to our equation, we get:

log₇[(3x³ + x) / x] = 2

Next, we simplify the expression inside the logarithm by dividing both terms in the numerator by x:

log₇(3x² + 1) = 2

Now, we convert the logarithmic equation to its exponential form. Recall that logₐ(b) = c is equivalent to aᶜ = b. Applying this to our equation, we get:

7² = 3x² + 1

Simplifying, we have:

49 = 3x² + 1

Subtracting 1 from both sides, we get:

48 = 3x²

Dividing both sides by 3, we obtain:

16 = x²

Taking the square root of both sides, we arrive at two potential solutions:

x = ±4

Having obtained two potential solutions, x = 4 and x = -4, we must now exercise caution and verify whether these solutions are genuine or extraneous. To do this, we substitute each solution back into the original logarithmic equation, log₇(3x³+x) - log₇(x) = 2, and check if it holds true.

Let us first consider x = 4. Substituting this value into the original equation, we get:

log₇(3(4)³ + 4) - log₇(4) = 2

Simplifying, we have:

log₇(192 + 4) - log₇(4) = 2

log₇(196) - log₇(4) = 2

Applying the logarithmic property of subtraction, we get:

log₇(196/4) = 2

log₇(49) = 2

Since 7² = 49, the equation holds true for x = 4. Therefore, x = 4 is a valid solution.

Now, let us examine x = -4. Substituting this value into the original equation, we get:

log₇(3(-4)³ + (-4)) - log₇(-4) = 2

Simplifying, we have:

log₇(-192 - 4) - log₇(-4) = 2

log₇(-196) - log₇(-4) = 2

Here, we encounter a critical issue. The logarithms of negative numbers are undefined. Therefore, x = -4 results in undefined terms in the equation, making it an extraneous solution. It is a mathematical illusion that arises from the algebraic manipulations but does not satisfy the fundamental restrictions of logarithmic functions.

Through our meticulous analysis, we have identified x = -4 as the extraneous solution to the logarithmic equation log₇(3x³+x) - log₇(x) = 2. This solution emerged during the solving process but does not hold true when substituted back into the original equation due to the presence of logarithms of negative numbers. The only valid solution to the equation is x = 4.

Therefore, among the given options:

A. x = -16

B. x = -4

C. x = 4

D. x = 16

The correct answer is B. x = -4, as it represents the extraneous solution.

Solving logarithmic equations can be a rewarding yet challenging endeavor. To navigate this terrain successfully, it is essential to grasp the fundamental properties of logarithms, develop a systematic approach, and, most importantly, remain vigilant for the emergence of extraneous solutions. Here are some key takeaways and best practices to guide you in your quest for accurate solutions:

1. Master the Properties of Logarithms: Familiarize yourself with the fundamental properties of logarithms, such as the product rule, quotient rule, power rule, and change-of-base formula. These properties serve as the essential tools for simplifying logarithmic expressions and equations.

2. Convert to Exponential Form: When solving logarithmic equations, converting them to their equivalent exponential form is often a crucial step. This transformation allows you to eliminate the logarithms and work with algebraic expressions, making the equation more amenable to solving.

3. Isolate the Logarithmic Terms: Before converting to exponential form, strive to isolate the logarithmic terms on one side of the equation. This often involves combining logarithmic terms using the properties of logarithms or performing algebraic manipulations to separate the logarithmic terms from other expressions.

4. Solve the Resulting Equation: Once you have converted the logarithmic equation to an algebraic form, employ the appropriate algebraic techniques to solve the resulting equation. This may involve solving linear equations, quadratic equations, or other types of algebraic equations.

5. Verify Your Solutions: The most crucial step in solving logarithmic equations is to verify your solutions. Substitute each potential solution back into the original logarithmic equation and check if it holds true. Any solution that results in undefined terms, such as logarithms of negative numbers or zero, is an extraneous solution and must be discarded.

6. Pay Attention to the Domain: Always be mindful of the domain of logarithmic functions. The argument of a logarithm must be positive. Therefore, any solution that makes the argument of a logarithm non-positive is an extraneous solution.

7. Be Meticulous and Organized: Solving logarithmic equations requires careful attention to detail and a systematic approach. Keep your work organized, clearly show each step, and double-check your calculations to minimize the risk of errors.

By adhering to these guidelines and cultivating a meticulous approach, you can confidently navigate the world of logarithmic equations, identify and eliminate extraneous solutions, and arrive at the genuine solutions that satisfy the equations.

In conclusion, solving logarithmic equations demands a blend of algebraic prowess and a keen awareness of the domain restrictions imposed by logarithmic functions. Extraneous solutions, those mathematical phantoms that emerge during the solving process but fail to satisfy the original equation, can pose a significant challenge. By mastering the properties of logarithms, employing a systematic approach, and diligently verifying solutions, we can effectively identify and eliminate extraneous solutions, ensuring we arrive at the accurate answers. The equation log₇(3x³+x) - log₇(x) = 2 serves as a compelling illustration of this concept, highlighting the importance of meticulous verification and a deep understanding of the underlying principles of logarithmic equations. Remember, in the realm of mathematics, accuracy and rigor are paramount, and the pursuit of genuine solutions requires a vigilant eye and a steadfast commitment to the fundamental principles.