What Is The Solution To The Equation Log₂(2x³ - 8) - 2log₂x = Log₂x?

In the realm of mathematics, logarithmic equations often present intriguing challenges. These equations, which involve logarithms, require a solid understanding of logarithmic properties and algebraic manipulation to solve effectively. In this comprehensive exploration, we will dissect the logarithmic equation log₂(2x³ - 8) - 2log₂x = log₂x, employing a step-by-step approach to unravel its intricacies and arrive at the correct solution. We'll delve into the fundamental principles of logarithms, including the power rule, quotient rule, and the critical concept of the domain of logarithmic functions. Furthermore, we'll navigate the algebraic techniques necessary to simplify the equation and isolate the variable, ultimately leading us to the solution set. This journey will not only provide the answer to the equation but also deepen your understanding of logarithmic problem-solving strategies.

Before we dive into the specifics of solving the equation, it's crucial to establish a strong foundation in the basics of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, 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, log₂8 = 3 because 2 raised to the power of 3 equals 8. Understanding this fundamental relationship between logarithms and exponents is key to manipulating and solving logarithmic equations. Additionally, we must consider the domain of logarithmic functions. The argument of a logarithm (the expression inside the logarithm) must always be positive. This restriction arises from the fact that logarithms are only defined for positive numbers. We'll need to keep this domain restriction in mind as we solve the equation to ensure that our solutions are valid. With these foundational concepts in place, we're well-equipped to embark on the journey of solving log₂(2x³ - 8) - 2log₂x = log₂x.

Unveiling the Solution: A Step-by-Step Approach

Our initial equation is log₂(2x³ - 8) - 2log₂x = log₂x. The first step in solving this equation is to consolidate the logarithmic terms. We can achieve this by utilizing the power rule of logarithms, which states that logₐ(bⁿ) = nlogₐ(b). Applying this rule to the term 2log₂x, we can rewrite it as log₂(x²). This transformation simplifies the equation, making it easier to manipulate. The equation now becomes log₂(2x³ - 8) - log₂(x²) = log₂x. The next step involves employing the quotient rule of logarithms. This rule states that logₐ(b) - logₐ(c) = logₐ(b/c). By applying this rule to the left side of the equation, we can combine the two logarithmic terms into a single term. This significantly simplifies the equation and brings us closer to isolating the variable x. After applying the quotient rule, our equation transforms into log₂((2x³ - 8)/x²) = log₂x. This simplified form allows us to eliminate the logarithms, a crucial step in solving for x.

Now that we have a single logarithm on each side of the equation, we can leverage the fundamental property of logarithms: if logₐ(b) = logₐ(c), then b = c. Applying this property to our equation, log₂((2x³ - 8)/x²) = log₂x, we can eliminate the logarithms and obtain the algebraic equation (2x³ - 8)/x² = x. This equation is now free of logarithms and can be solved using standard algebraic techniques. However, before we proceed with solving for x, it's essential to address the domain restriction of logarithmic functions. As mentioned earlier, the argument of a logarithm must be positive. Therefore, we need to ensure that both 2x³ - 8 and x are greater than zero. This restriction will help us identify any extraneous solutions that may arise during the algebraic manipulation. With the domain restriction in mind, we can now confidently proceed to solve the algebraic equation and determine the valid solutions for x. The next steps will involve simplifying the equation, isolating x, and verifying the solutions against the domain restriction.

Algebraic Gymnastics: Solving for x

Having arrived at the algebraic equation (2x³ - 8)/x² = x, our next task is to solve for x. The first step in this process is to eliminate the fraction. We can achieve this by multiplying both sides of the equation by x². This operation yields 2x³ - 8 = x³. Next, we aim to consolidate the terms involving x. Subtracting x³ from both sides of the equation, we obtain x³ - 8 = 0. This equation is a difference of cubes, a special algebraic form that can be factored using the identity a³ - b³ = (a - b)(a² + ab + b²). Applying this factorization to our equation, with a = x and b = 2, we get (x - 2)(x² + 2x + 4) = 0. This factored form provides us with two potential solutions for x. The first solution arises from the factor (x - 2), which gives us x = 2. The second potential solutions come from the quadratic factor (x² + 2x + 4). To find these solutions, we can use the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions are given by x = (-b ± √(b² - 4ac)) / (2a). Applying this formula to our quadratic factor, we find that the discriminant (b² - 4ac) is negative. This indicates that the quadratic factor has no real roots. Therefore, the only real solution to our equation is x = 2.

However, our journey is not yet complete. We must now verify this solution against the domain restriction we established earlier. Recall that the argument of a logarithm must be positive. In our original equation, log₂(2x³ - 8) - 2log₂x = log₂x, we have two logarithmic terms with arguments 2x³ - 8 and x. We need to ensure that both of these expressions are positive when x = 2. Substituting x = 2 into 2x³ - 8, we get 2(2³) - 8 = 2(8) - 8 = 16 - 8 = 8, which is positive. Similarly, substituting x = 2 into x, we get 2, which is also positive. Therefore, x = 2 satisfies the domain restriction and is a valid solution to the logarithmic equation. We have successfully navigated the algebraic manipulations and domain considerations to arrive at the final solution. The solution x = 2 not only satisfies the algebraic equation but also adheres to the fundamental principles of logarithms, making it the definitive answer to our problem. With this solution in hand, we can confidently conclude our exploration of this logarithmic equation.

The Verdict: The Solution Unveiled

After a meticulous journey through logarithmic properties, algebraic manipulations, and domain restrictions, we have arrived at the solution to the equation log₂(2x³ - 8) - 2log₂x = log₂x. Our step-by-step approach involved consolidating logarithmic terms using the power rule and quotient rule, eliminating logarithms by equating arguments, solving the resulting algebraic equation, and crucially, verifying the solution against the domain restriction of logarithmic functions. This rigorous process has led us to the definitive answer: x = 2. This solution not only satisfies the algebraic equation but also adheres to the fundamental requirement that the arguments of logarithms must be positive. The solution stands as a testament to the power of combining logarithmic principles with algebraic techniques to unravel complex mathematical problems. By understanding the underlying concepts and applying them methodically, we can navigate the challenges posed by logarithmic equations and arrive at accurate solutions.

Final Answer:

The final answer is B. x = 2.

Decoding Logarithmic Equations: Frequently Asked Questions

What is a logarithmic equation?

A logarithmic equation is an equation that includes a logarithm of an expression containing a variable. Solving these equations often involves using the properties of logarithms to isolate the variable. Logarithms 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 example, if we have the equation log₂(8) = x, this means 2 raised to the power of x equals 8. Thus, x = 3 because 2³ = 8. Understanding this basic relationship is crucial for manipulating and solving logarithmic equations.

What are the key properties of logarithms used in solving equations?

Several key properties of logarithms are essential when solving logarithmic equations. These include:

• Product Rule: logₐ(mn) = logₐ(m) + logₐ(n). This rule states that the logarithm of the product of two numbers is the sum of their logarithms.
• Quotient Rule: logₐ(m/n) = logₐ(m) - logₐ(n). This rule states that the logarithm of the quotient of two numbers is the difference of their logarithms.
• Power Rule: logₐ(mⁿ) = nlogₐ(m). This rule allows us to move exponents from the argument of the logarithm to the front as a coefficient.
• Change of Base Rule: log𝑏(a) = logₓ(a) / logₓ(b). This rule allows us to change the base of a logarithm, which is particularly useful when using calculators that only have common (base 10) and natural (base e) logarithms.
• Inverse Property: a^(logₐ(x)) = x and logₐ(aˣ) = x. These properties highlight the inverse relationship between logarithms and exponentiation.

These properties enable us to manipulate logarithmic equations, combine or separate terms, and ultimately isolate the variable we are solving for. Mastering these properties is key to successfully tackling logarithmic problems.

Why is it important to check solutions in logarithmic equations?

Checking solutions in logarithmic equations is crucial because the domain of logarithmic functions is restricted to positive numbers. The argument of a logarithm (the expression inside the logarithm) must always be greater than zero. This restriction arises from the fact that logarithms are only defined for positive numbers. When solving logarithmic equations, we may obtain solutions that, when substituted back into the original equation, result in taking the logarithm of a non-positive number. These solutions are called extraneous solutions and are not valid. Therefore, it is essential to check all potential solutions by substituting them back into the original equation and ensuring that the argument of every logarithm is positive. This step ensures that the solutions we obtain are mathematically valid and consistent with the domain of logarithmic functions.

What is an extraneous solution in the context of logarithmic equations?

An extraneous solution in the context of logarithmic equations is a value that satisfies the transformed equation but does not satisfy the original logarithmic equation. This typically occurs because the domain of logarithmic functions is restricted to positive numbers. When solving logarithmic equations, we often perform algebraic manipulations, such as raising both sides to a power or combining logarithmic terms, which can introduce solutions that are not valid in the original equation. These invalid solutions arise when the potential solution, when substituted back into the original equation, results in taking the logarithm of a non-positive number (zero or negative). Since the logarithm of a non-positive number is undefined, these solutions are extraneous and must be discarded. Checking for extraneous solutions is a critical step in solving logarithmic equations to ensure that the final answer is mathematically correct and consistent with the domain of logarithmic functions.

How do you solve a logarithmic equation with different bases?

Solving a logarithmic equation with different bases requires a preliminary step of converting all logarithms to a common base. This is typically achieved using the change of base rule, which states that log𝑏(a) = logₓ(a) / logₓ(b), where x is the new base. The choice of the new base is arbitrary, but common choices include base 10 (common logarithm) or base e (natural logarithm) because most calculators have functions for these bases. Once all logarithms are in the same base, you can then proceed to simplify the equation using the properties of logarithms, such as the product rule, quotient rule, and power rule. After simplification, you can eliminate the logarithms by equating arguments or by using exponentiation, depending on the structure of the equation. Finally, it is essential to check the solutions against the domain restriction of logarithmic functions to identify and discard any extraneous solutions. This process ensures that the final solutions are mathematically valid and consistent with the original equation.