Solving The Logarithmic Equation 2 Log₂x - Log₂(2x) = 3
In the realm of mathematics, particularly when dealing with logarithmic equations, it's crucial to understand the underlying principles and properties that govern these expressions. This article delves into solving the equation 2 log₂x - log₂(2x) = 3, providing a step-by-step solution and explanation to enhance your understanding of logarithmic manipulations. Our goal is to transform the given equation into a manageable form, isolating the variable x and ultimately determining its value. The process involves employing key logarithmic identities, such as the power rule and the quotient rule, to simplify the equation. By meticulously applying these rules, we can reduce the complexity of the equation and pave the way for a straightforward algebraic solution. This exploration is not just about finding the answer; it's about fostering a deeper comprehension of how logarithms work and how they can be used to solve various mathematical problems. So, let's embark on this journey of logarithmic exploration and unravel the solution to the equation at hand.
Understanding Logarithmic Properties
Before diving into the solution, it's essential to grasp the fundamental logarithmic properties that will be our tools in this endeavor. These properties act as the building blocks for manipulating and simplifying logarithmic expressions. The two key properties we'll utilize are the power rule and the quotient rule.
Power Rule
The power rule of logarithms states that logₐ(bᶜ) = c logₐ(b). This rule allows us to move exponents within a logarithm as coefficients outside the logarithm, and vice versa. In simpler terms, if you have a logarithm of a number raised to a power, you can bring the power down and multiply it by the logarithm of the number. This is incredibly useful for simplifying expressions where the variable x might be trapped within an exponent or a logarithmic term. For instance, in our equation, the term 2 log₂x can be seen as a direct application of this rule in reverse, as we might want to rewrite it as log₂(x²) at some point. Understanding this interplay between exponents and coefficients is crucial for effectively manipulating logarithmic equations. The power rule is not just a mathematical trick; it reflects the fundamental relationship between exponents and logarithms, which are inverse operations of each other. The ability to seamlessly move between these forms is a hallmark of proficiency in logarithmic problem-solving.
Quotient Rule
The quotient rule states that logₐ(b/c) = logₐ(b) - logₐ(c). This rule allows us to separate the logarithm of a quotient into the difference of two logarithms. Conversely, the difference of two logarithms with the same base can be combined into a single logarithm of a quotient. This property is particularly useful when we have terms like log₂(2x), which involves the logarithm of a product. We can't directly apply the quotient rule here, but understanding that it's related to the logarithm of a division is important. The quotient rule helps us break down complex logarithmic expressions into simpler, more manageable parts. It's a testament to the inherent structure of logarithms, which mirrors the properties of exponents. Just as dividing numbers with the same base involves subtracting their exponents, taking the logarithm of a quotient involves subtracting the logarithms of the individual terms. Mastering this rule is essential for simplifying and solving a wide range of logarithmic equations.
Step-by-Step Solution
Now, let's apply these properties to solve the equation 2 log₂x - log₂(2x) = 3. We'll proceed systematically, breaking down each step for clarity.
Step 1: Applying the Power Rule
Our first move is to apply the power rule to the term 2 log₂x. Using the rule c logₐ(b) = logₐ(bᶜ), we can rewrite this term as log₂(x²). This transformation is a crucial step in simplifying the equation, as it allows us to consolidate the logarithmic terms. By bringing the coefficient 2 inside the logarithm as an exponent, we effectively reduce the number of logarithmic terms in the equation. This might seem like a small change, but it sets the stage for further simplification. The power rule, in this context, acts as a bridge between coefficients and exponents, allowing us to manipulate the equation in a way that brings us closer to the solution. It's a powerful tool in our logarithmic arsenal, enabling us to reshape expressions and reveal hidden relationships between variables and constants. Understanding how to apply this rule in various contexts is essential for mastering logarithmic equations.
Our equation now looks like this: log₂(x²) - log₂(2x) = 3.
Step 2: Applying the Quotient Rule
Next, we'll apply the quotient rule to combine the two logarithmic terms on the left-hand side. The quotient rule, logₐ(b) - logₐ(c) = logₐ(b/c), allows us to combine the difference of two logarithms into a single logarithm of a quotient. In our case, we have log₂(x²) - log₂(2x), which can be rewritten as log₂(x²/2x). This step significantly simplifies the equation, as we've reduced two logarithmic terms into one. The quotient rule acts as a unifying force, bringing together separate logarithmic expressions into a single, more concise term. This simplification is not just about reducing the number of terms; it's about revealing the underlying structure of the equation. By combining the logarithms, we expose the relationship between the arguments of the logarithms, in this case, x² and 2x. This unified expression is much easier to manipulate and solve, bringing us closer to isolating the variable x. The quotient rule is a testament to the elegance and efficiency of logarithmic properties, allowing us to condense complex expressions into manageable forms.
Simplifying the fraction inside the logarithm, x²/2x becomes x/2. So, our equation now reads: log₂(x/2) = 3.
Step 3: Converting to Exponential Form
To eliminate the logarithm, we need to convert the equation from logarithmic form to exponential form. The logarithmic equation logₐ(b) = c is equivalent to the exponential equation aᶜ = b. In our case, log₂(x/2) = 3 can be rewritten as 2³ = x/2. This transformation is the key to unlocking the variable x from the confines of the logarithm. Converting to exponential form allows us to work with a more familiar algebraic equation, one that doesn't involve logarithmic functions. It's a fundamental technique in solving logarithmic equations, acting as a bridge between two different mathematical representations of the same relationship. The exponential form highlights the inverse relationship between logarithms and exponents, showcasing how they can be used to undo each other. This step is not just a mechanical process; it's a conceptual leap that allows us to see the equation in a new light, paving the way for a straightforward algebraic solution. Understanding this conversion is crucial for mastering logarithmic equations and their applications.
Step 4: Solving for x
Now we have a simple algebraic equation: 2³ = x/2. We know that 2³ = 8, so the equation becomes 8 = x/2. To isolate x, we multiply both sides of the equation by 2. This gives us x = 16. This final step is the culmination of our logarithmic journey, the point where we finally unveil the value of the unknown variable. It's a testament to the power of algebraic manipulation, the ability to transform equations and isolate variables through a series of logical steps. The solution x = 16 is not just a numerical answer; it's the result of a systematic application of logarithmic properties and algebraic techniques. It represents the specific value of x that satisfies the original equation, the point where the logarithmic relationship holds true. This final step underscores the importance of precision and accuracy in mathematical problem-solving, ensuring that each step leads us closer to the correct solution. The value x = 16 is the reward for our efforts, the answer we sought from the beginning.
The Answer
Therefore, the solution to the equation 2 log₂x - log₂(2x) = 3 is x = 16. Looking at the options provided:
A. x = 16 (Correct) B. x = 18 C. x = 6 D. x = 8
The correct answer is A. x = 16. This answer is the fruit of our labor, the result of carefully applying logarithmic properties and algebraic techniques. It's not just a random selection from the options; it's the value that satisfies the original equation, the point where the logarithmic relationship holds true. The process of arriving at this answer has been a journey of mathematical exploration, a demonstration of the power and elegance of logarithmic functions. The correct answer stands as a testament to our understanding of these principles, a validation of our ability to manipulate equations and solve for unknowns. It's a moment of satisfaction, the culmination of a step-by-step process that has led us to the truth.
Conclusion
In conclusion, solving the logarithmic equation 2 log₂x - log₂(2x) = 3 requires a solid understanding of logarithmic properties and their applications. By systematically applying the power rule and the quotient rule, we were able to simplify the equation and ultimately find the solution, x = 16. This exercise highlights the importance of mastering these fundamental principles in mathematics. The journey through this equation has been a demonstration of the power of mathematical reasoning, the ability to transform complex problems into manageable steps. It's a testament to the elegance of logarithmic functions, their ability to describe relationships and solve for unknowns. The final solution, x = 16, is not just a numerical answer; it's a symbol of our understanding, a validation of our skills in logarithmic manipulation. This exploration serves as a reminder that mathematics is not just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively to solve problems. The ability to navigate logarithmic equations is a valuable skill, one that opens doors to further mathematical exploration and application in various fields.
