Solving Logarithmic Equations Log₂(x + 6) = 3 - Log₂(x + 4)

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In the realm of mathematics, logarithmic equations often present a unique challenge, demanding a solid understanding of logarithmic properties and algebraic manipulation. Our focus today is on unraveling the complexities of such equations, specifically addressing the equation log₂(x + 6) = 3 - log₂(x + 4). We will embark on a step-by-step journey, transforming this equation into a solvable form, exploring potential solutions, and rigorously verifying their validity. This comprehensive guide aims not only to provide the solution but also to enhance your problem-solving skills in the broader context of logarithmic equations.

Understanding Logarithms: The Foundation of Our Solution

Before we dive into the specifics of the equation at hand, it's crucial to lay a strong foundation by revisiting the fundamentals of logarithms. A logarithm, at its core, is the inverse operation to exponentiation. The expression logₐ(b) = c essentially asks: “To what power must we raise a (the base) to obtain b?” The answer, of course, is c. Understanding this relationship is paramount to manipulating and solving logarithmic equations effectively.

In the given equation, log₂(x + 6) = 3 - log₂(x + 4), the base of the logarithm is 2. This means we are dealing with powers of 2. To solve this equation, we will leverage key properties of logarithms, such as the product rule, quotient rule, and the power rule. These rules allow us to combine and simplify logarithmic expressions, paving the way for a solution. Furthermore, we must be mindful of the domain of logarithmic functions. Logarithms are only defined for positive arguments, meaning that the expressions inside the logarithm must be greater than zero. This constraint will play a critical role in verifying the validity of our solutions.

Step-by-Step Solution: Unraveling the Equation

Now, let's tackle the equation log₂(x + 6) = 3 - log₂(x + 4) head-on. Our strategy involves isolating the logarithmic terms, combining them using logarithmic properties, and then converting the equation into exponential form. By following these steps meticulously, we can systematically work towards a solution.

  1. Isolate Logarithmic Terms: The first step in our journey is to gather all the logarithmic terms on one side of the equation. To achieve this, we add log₂(x + 4) to both sides of the equation:

    log₂(x + 6) + log₂(x + 4) = 3

    This maneuver brings the logarithmic terms together, setting the stage for the next step.

  2. Apply the Product Rule: The product rule of logarithms states that the logarithm of the product of two numbers is equal to the sum of the logarithms of those numbers. Mathematically, this is expressed as:

    logₐ(m) + logₐ(n) = logₐ(mn)

    Applying this rule to our equation, we combine the two logarithmic terms on the left side:

    log₂((x + 6)(x + 4)) = 3

    This simplification significantly reduces the complexity of the equation.

  3. Convert to Exponential Form: To eliminate the logarithm, we convert the equation from logarithmic form to exponential form. Recall that logₐ(b) = c is equivalent to aᶜ = b. Applying this to our equation, we get:

    2³ = (x + 6)(x + 4)

    This transformation removes the logarithm, leaving us with a polynomial equation.

  4. Simplify and Solve the Quadratic Equation: Now, we simplify and solve the resulting quadratic equation. First, we expand the right side and simplify the left side:

    8 = x² + 10x + 24

    Next, we rearrange the equation to bring all terms to one side:

    x² + 10x + 16 = 0

    This is a standard quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We look for two numbers that multiply to 16 and add up to 10. These numbers are 2 and 8. Thus, we can factor the quadratic as:

    (x + 2)(x + 8) = 0

    This gives us two potential solutions:

    x = -2 or x = -8

  5. Verify the Solutions: An essential step in solving logarithmic equations is to verify the solutions. Remember that the arguments of the logarithms must be positive. We need to check if our solutions, x = -2 and x = -8, satisfy this condition in the original equation.

    For x = -2:

    x + 6 = -2 + 6 = 4 > 0

    x + 4 = -2 + 4 = 2 > 0

    Both arguments are positive, so x = -2 is a valid solution.

    For x = -8:

    x + 6 = -8 + 6 = -2 < 0

    x + 4 = -8 + 4 = -4 < 0

    Both arguments are negative, so x = -8 is not a valid solution. It is an extraneous solution that arises from the algebraic manipulation but does not satisfy the original equation's domain restrictions.

Final Answer: The Solution to the Equation

After careful verification, we conclude that the only valid solution to the equation log₂(x + 6) = 3 - log₂(x + 4) is x = -2. This solution satisfies the original equation and adheres to the domain restrictions of logarithmic functions. The extraneous solution, x = -8, highlights the importance of verifying solutions in the context of logarithmic equations.

Key Concepts and Takeaways

Solving logarithmic equations requires a firm grasp of logarithmic properties, algebraic techniques, and the concept of domain restrictions. In this comprehensive guide, we have demonstrated a systematic approach to solving the equation log₂(x + 6) = 3 - log₂(x + 4). We isolated logarithmic terms, applied the product rule, converted the equation to exponential form, solved the resulting quadratic equation, and, crucially, verified the solutions.

Here are some key takeaways from our exploration:

  • Logarithmic Properties: The product rule, quotient rule, and power rule are essential tools for manipulating logarithmic expressions.
  • Domain Restrictions: Logarithms are only defined for positive arguments. Always check solutions to ensure they satisfy this condition.
  • Extraneous Solutions: Algebraic manipulations can sometimes introduce extraneous solutions. Verification is crucial to identify and discard these solutions.
  • Systematic Approach: A structured approach, involving isolating terms, applying properties, and converting forms, is key to solving logarithmic equations effectively.

By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of logarithmic equations and enhance your problem-solving abilities in mathematics.

Practice Problems: Sharpening Your Skills

To solidify your understanding of solving logarithmic equations, consider tackling the following practice problems:

  1. Solve for x: log₃(2x + 1) = 2 - log₃(x - 1)
  2. Solve for x: log₄(x + 2) + log₄(x - 1) = 1
  3. Solve for x: 2log₅(x) = log₅(2x + 3)

Working through these problems will provide valuable practice and reinforce the concepts discussed in this guide. Remember to apply the same systematic approach and always verify your solutions.

Conclusion: Mastering Logarithmic Equations

In conclusion, solving logarithmic equations is a skill that combines algebraic proficiency with a deep understanding of logarithmic properties and domain restrictions. By following a systematic approach, verifying solutions, and practicing consistently, you can master this essential mathematical skill. The journey through the equation log₂(x + 6) = 3 - log₂(x + 4) serves as a valuable example, illustrating the key steps and considerations involved in solving logarithmic equations. Keep exploring, keep practicing, and you will undoubtedly excel in the realm of logarithms and beyond.