Solving Logarithmic Equations: A Step-by-Step Guide

In the realm of mathematics, logarithmic equations often present a unique challenge, requiring a blend of algebraic manipulation and a deep understanding of logarithmic properties. Here, we embark on a journey to unravel the solution of a specific logarithmic equation: log₈ 16 + 2 log₈ x = 2. To effectively navigate this equation, we'll meticulously dissect each component, leveraging the fundamental principles of logarithms to arrive at the correct answer.

Deciphering the Logarithmic Puzzle: A Step-by-Step Approach

Our quest begins with a careful examination of the equation itself: log₈ 16 + 2 log₈ x = 2. This equation presents us with two logarithmic terms, each possessing its own unique characteristics. The first term, log₈ 16, represents the power to which we must raise the base 8 to obtain the value 16. The second term, 2 log₈ x, introduces a variable, x, within the logarithmic expression, adding another layer of complexity to our solution-seeking endeavor.

To effectively tackle this equation, we must first familiarize ourselves with the fundamental properties of logarithms. These properties serve as the bedrock of our solution strategy, allowing us to manipulate and simplify the equation into a more manageable form. Among these properties, the power rule of logarithms stands out as particularly relevant to our current task. This rule states that logₐ (bᶜ) = c logₐ b, where a, b, and c are constants. In essence, the power rule allows us to move exponents within a logarithmic expression, simplifying the equation and paving the way for further manipulation.

Applying the power rule to the second term in our equation, 2 log₈ x, we can rewrite it as log₈ (x²). This transformation allows us to consolidate the two logarithmic terms into a single expression, making the equation more amenable to simplification. Our equation now takes on a new form: log₈ 16 + log₈ (x²) = 2. With this crucial step accomplished, we can proceed to leverage another fundamental property of logarithms: the product rule. This rule states that logₐ b + logₐ c = logₐ (b * c), where a, b, and c are constants. The product rule empowers us to combine two logarithmic terms with the same base into a single logarithmic term, further simplifying our equation.

By applying the product rule to our transformed equation, we can combine the two logarithmic terms on the left-hand side: log₈ 16 + log₈ (x²) = log₈ (16 * x²). Our equation now stands as: log₈ (16x²) = 2. With the logarithmic terms consolidated, we are poised to eliminate the logarithm altogether, paving the way for solving for x.

Eradicating the Logarithm: Unleashing the Variable

To eliminate the logarithm, we must recall the fundamental definition of a logarithm. The expression logₐ b = c is equivalent to the exponential expression aᶜ = b. This equivalence allows us to transform our logarithmic equation into an exponential one, effectively banishing the logarithm and liberating the variable x.

Applying this principle to our equation, log₈ (16x²) = 2, we can rewrite it in exponential form as 8² = 16x². This transformation brings us closer to our goal of isolating x and determining its value. Simplifying the exponential term, we have 64 = 16x². Now, our equation takes on a more familiar algebraic form, allowing us to employ standard algebraic techniques to solve for x.

To isolate x², we divide both sides of the equation by 16, resulting in 4 = x². At this juncture, we encounter a crucial point to consider: the square root property. This property states that if x² = a, then x = ±√a. Applying this property to our equation, we obtain x = ±√4. Evaluating the square root, we arrive at two potential solutions: x = 2 and x = -2.

However, we must exercise caution when dealing with logarithmic equations. Logarithms are only defined for positive arguments. This means that the value inside the logarithm must be strictly greater than zero. Therefore, we must examine our potential solutions to ensure they satisfy this condition.

Verifying the Solutions: Ensuring Logarithmic Harmony

Substituting x = -2 back into the original equation, we encounter a problem: log₈ (-2) is undefined, as the logarithm of a negative number does not exist. Consequently, we must discard x = -2 as an extraneous solution, one that arises from our algebraic manipulations but does not satisfy the original equation. On the other hand, substituting x = 2 into the original equation, we find that it holds true. Therefore, x = 2 is the sole valid solution to our logarithmic equation.

In conclusion, through a meticulous application of logarithmic properties and algebraic techniques, we have successfully navigated the intricacies of the equation log₈ 16 + 2 log₈ x = 2. Our journey has led us to the definitive solution: x = 2. This solution stands as a testament to the power of mathematical reasoning and the importance of verifying solutions within the context of the original problem.

The correct answer is x = 2. This was determined by applying the power rule and product rule of logarithms, converting the logarithmic equation to an exponential equation, and solving for x. It's essential to verify the solutions in the original equation due to the domain restrictions of logarithms.

Keywords: Logarithmic equations, power rule of logarithms, product rule of logarithms, solving logarithmic equations, extraneous solutions, domain restrictions of logarithms, mathematical reasoning, algebraic techniques.

Solving Logarithmic Equations Step-by-Step: A Detailed Explanation