Solving Logarithmic Equations: A Step-by-Step Guide

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Introduction

In this comprehensive exploration, we will delve into the intricacies of solving the logarithmic equation log₂(2x³ - 8) - 2log₂x = log₂x. This equation, a blend of logarithmic functions and algebraic expressions, presents an engaging challenge for mathematics enthusiasts. Our discussion will not only pinpoint the solution but also illuminate the underlying principles of logarithmic manipulations and equation-solving strategies. This journey is tailored for students, educators, and anyone with an interest in mathematics, ensuring a clear and thorough understanding of the process. Our primary goal is to provide a detailed, step-by-step solution while enriching the reader's grasp of logarithmic equations. By the end of this exploration, you will not only know the answer but also appreciate the mathematical elegance involved in solving such problems. So, let's embark on this mathematical quest to unravel the mysteries of this equation and enhance our problem-solving prowess.

Understanding Logarithmic Equations

Before we dive into the specifics of our equation, it's crucial to lay a strong foundation in logarithmic principles. Logarithmic equations, at their core, are expressions that involve logarithms. A logarithm is essentially the inverse operation of exponentiation. To put it simply, 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, log₂8 = 3 because 2 raised to the power of 3 equals 8. Understanding this fundamental relationship is key to tackling logarithmic equations effectively. Logarithmic equations come in various forms, ranging from simple expressions to complex combinations of logarithms and algebraic terms, just like the one we are about to solve: log₂(2x³ - 8) - 2log₂x = log₂x.

Properties of Logarithms are the building blocks for simplifying and solving these equations. These properties allow us to manipulate logarithmic expressions into more manageable forms. Some of the most important properties include:

  1. Product Rule: logₐ(mn) = logₐm + logₐn
  2. Quotient Rule: logₐ(m/n) = logₐm - logₐn
  3. Power Rule: logₐ(mᵖ) = p * logₐm
  4. Change of Base Rule: logₐb = logₓb / logₓa

In our equation, the Power Rule and the Quotient Rule will be particularly useful. Recognizing when and how to apply these properties is a crucial skill in solving logarithmic equations. Another vital aspect of working with logarithms is understanding their domain. Logarithms are only defined for positive arguments. This means that when solving a logarithmic equation, we must ensure that the solutions we obtain do not result in taking the logarithm of a non-positive number. This often involves checking our solutions against the original equation to rule out extraneous roots. With these principles in mind, we are well-equipped to tackle the equation at hand. Let's proceed by applying these foundational concepts to unravel the solution of log₂(2x³ - 8) - 2log₂x = log₂x. The journey of solving logarithmic equations is not just about finding the answer; it's about deepening our understanding of mathematical relationships and honing our problem-solving skills.

Detailed Solution of the Equation

Now, let's embark on the journey of solving the logarithmic equation log₂(2x³ - 8) - 2log₂x = log₂x step-by-step. Our goal is to isolate x by skillfully applying logarithmic properties and algebraic techniques.

Step 1: Simplify the Equation

The first step in solving this equation is to simplify it using the properties of logarithms. Notice the term 2log₂x. We can use the Power Rule of logarithms, which states that logₐ(mᵖ) = p * logₐm, to rewrite this term. Applying the Power Rule, we get:

  • 2log₂x = log₂(x²)

Substituting this back into our original equation, we have:

  • log₂(2xÂł - 8) - log₂(x²) = log₂x

Next, we can use the Quotient Rule of logarithms, which states that logₐ(m/n) = logₐm - logₐn. Applying this rule to the left side of the equation, we combine the two logarithmic terms:

  • log₂((2xÂł - 8) / x²) = log₂x

This simplification is a crucial step as it consolidates the logarithmic terms, making the equation easier to handle.

Step 2: Remove the Logarithms

Now that we have a single logarithm on each side of the equation, we can eliminate the logarithms. Since the bases of the logarithms are the same (base 2), we can equate the arguments of the logarithms. This is because if logₐm = logₐn, then m = n. Applying this principle, we get:

  • (2xÂł - 8) / x² = x

This step transforms the logarithmic equation into an algebraic equation, which we can solve using standard algebraic techniques.

Step 3: Solve the Algebraic Equation

To solve the algebraic equation (2x³ - 8) / x² = x, we first want to eliminate the fraction. We can do this by multiplying both sides of the equation by x²:

  • x² * ((2xÂł - 8) / x²) = x * x²

This simplifies to:

  • 2xÂł - 8 = xÂł

Now, we want to isolate the terms involving x. Subtract xÂł from both sides of the equation:

  • 2xÂł - xÂł - 8 = 0

This simplifies to:

  • xÂł - 8 = 0

This is a simple cubic equation. We can solve it by adding 8 to both sides:

  • xÂł = 8

To find x, we take the cube root of both sides:

  • x = ∛8

Thus, we find:

  • x = 2

Step 4: Check for Extraneous Solutions

It's crucial to check our solution in the original logarithmic equation to ensure it doesn't result in taking the logarithm of a non-positive number. Recall the original equation:

  • log₂(2xÂł - 8) - 2log₂x = log₂x

Substitute x = 2 into the equation:

  • log₂(2(2)Âł - 8) - 2log₂(2) = log₂(2)
  • log₂(16 - 8) - 2(1) = 1
  • log₂(8) - 2 = 1
  • 3 - 2 = 1
  • 1 = 1

Since the equation holds true and there are no logarithms of non-positive numbers, x = 2 is a valid solution.

Conclusion

Through a detailed, step-by-step approach, we have successfully solved the logarithmic equation log₂(2x³ - 8) - 2log₂x = log₂x. The solution we found is x = 2. This process involved applying key logarithmic properties such as the Power Rule and the Quotient Rule, simplifying the equation, and then solving the resulting algebraic equation. We also emphasized the importance of checking for extraneous solutions to ensure the validity of our answer. This exploration not only provides the solution to the equation but also reinforces the fundamental principles of logarithmic equations and problem-solving strategies. Understanding these principles is invaluable for anyone studying mathematics, as it enhances both their mathematical knowledge and their ability to tackle complex problems. The journey of solving this equation has been a testament to the power and elegance of mathematics. By breaking down a complex problem into manageable steps and applying the right tools and techniques, we can unravel the mysteries of mathematics and expand our understanding of the world around us.