Solving The Logarithmic Equation Log₂(x+2) + Log₂(x+1) = 1

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Introduction

In this article, we will delve into the process of solving a logarithmic equation. Logarithmic equations often appear complex, but with a systematic approach and understanding of logarithmic properties, they can be solved effectively. Here, we will tackle the equation log₂(x+2) + log₂(x+1) = 1, providing a detailed, step-by-step solution. This process not only helps in finding the value of x but also reinforces the application of logarithmic rules and algebraic manipulations. Understanding these techniques is crucial for anyone studying mathematics, especially in areas like calculus, where logarithmic functions are frequently encountered. The ability to confidently solve such equations is a valuable asset in both academic and practical contexts.

Understanding Logarithms

Before diving into the solution, let's briefly discuss logarithms. A logarithm is essentially the inverse operation to exponentiation. The expression logₐ(b) = c means that a raised to the power of c equals b. Here, a is the base of the logarithm, b is the argument, and c is the result. In our equation, the base is 2. Logarithms are incredibly useful in various fields, including computer science, physics, and engineering, for scaling down very large numbers or representing exponential relationships in a more manageable form. Familiarity with logarithmic properties, such as the product rule, quotient rule, and power rule, is essential for solving logarithmic equations. These rules allow us to simplify and manipulate logarithmic expressions, making them easier to solve. The most commonly used bases are 10 (common logarithm) and e (natural logarithm), but understanding logarithms with different bases, like the base 2 in our equation, is equally important.

Step-by-Step Solution

1. Combine Logarithms

The first step in solving log₂(x+2) + log₂(x+1) = 1 involves using the logarithmic product rule. This rule states that the sum of the logarithms of two numbers is equal to the logarithm of their product. In mathematical terms, logₐ(m) + logₐ(n) = logₐ(mn). Applying this rule to our equation, we combine the two logarithms on the left side: log₂((x+2)(x+1)) = 1. This simplification is crucial as it transforms the equation into a more manageable form. By combining the logarithms, we reduce the complexity of the equation, making it easier to proceed with the next steps. Understanding and applying logarithmic properties like the product rule is fundamental in solving logarithmic equations. This step demonstrates the power of these properties in simplifying complex expressions.

2. Convert to Exponential Form

Now that we have a single logarithm, we can convert the equation from logarithmic form to exponential form. The equation log₂((x+2)(x+1)) = 1 can be rewritten as 2¹ = (x+2)(x+1). This conversion is based on the fundamental definition of logarithms, which states that if logₐ(b) = c, then aᶜ = b. Converting to exponential form eliminates the logarithm, allowing us to work with a simpler algebraic equation. This step is a key technique in solving logarithmic equations, as it bridges the gap between logarithms and more familiar algebraic expressions. By understanding this conversion, one can easily transition between logarithmic and exponential forms, which is a critical skill in various mathematical contexts.

3. Expand and Simplify

Next, we need to expand the expression on the right side of the equation. Multiplying (x+2) and (x+1) gives us x² + 3x + 2. So, our equation becomes 2¹ = x² + 3x + 2, which simplifies to 2 = x² + 3x + 2. To solve this quadratic equation, we need to set it to zero. Subtracting 2 from both sides, we get 0 = x² + 3x. Expanding and simplifying the equation is a crucial step in transforming it into a standard form that we can easily solve. This process involves basic algebraic operations, such as multiplication and simplification, which are essential skills in mathematical problem-solving. By reducing the equation to a standard form, we can apply various techniques, such as factoring or using the quadratic formula, to find the solutions.

4. Solve the Quadratic Equation

The equation 0 = x² + 3x is a quadratic equation. We can solve it by factoring. Factoring out an x gives us x(x + 3) = 0. This means that either x = 0 or x + 3 = 0. Solving for x in each case, we find two potential solutions: x = 0 and x = -3. Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily factored. This method involves expressing the quadratic equation as a product of two linear factors. By setting each factor equal to zero, we can find the roots of the equation. Quadratic equations are common in mathematics and have applications in various fields, making it essential to master different solution techniques, such as factoring, completing the square, and using the quadratic formula.

5. Check for Extraneous Solutions

In logarithmic equations, it is crucial to check for extraneous solutions. These are solutions that satisfy the algebraic equation but do not work in the original logarithmic equation. This often happens because the logarithm of a negative number or zero is undefined. Let’s check our potential solutions, x = 0 and x = -3. For x = 0, the original equation becomes log₂(0+2) + log₂(0+1) = log₂(2) + log₂(1) = 1 + 0 = 1, which is true. For x = -3, we have log₂(-3+2) + log₂(-3+1) = log₂(-1) + log₂(-2). Since we cannot take the logarithm of a negative number, x = -3 is an extraneous solution. Checking for extraneous solutions is a critical step in solving logarithmic and radical equations. This step ensures that the solutions we obtain are valid within the domain of the original equation. Extraneous solutions can arise due to the nature of these types of equations, where certain operations can introduce solutions that do not satisfy the original equation's conditions.

Final Answer

After checking for extraneous solutions, we find that the only valid solution for the equation log₂(x+2) + log₂(x+1) = 1 is x = 0. The process of solving this equation involved several key steps, including combining logarithms, converting to exponential form, simplifying, solving the resulting algebraic equation, and, most importantly, checking for extraneous solutions. Each step highlights the importance of understanding logarithmic properties and algebraic techniques. This solution underscores the need for a thorough and careful approach when dealing with logarithmic equations, ensuring that all steps are logically sound and that the final answer is valid. The ability to solve logarithmic equations is a valuable skill in mathematics, with applications in various fields such as engineering, physics, and computer science.

Conclusion

Solving the logarithmic equation log₂(x+2) + log₂(x+1) = 1 demonstrates the importance of understanding logarithmic properties and algebraic manipulation. The step-by-step approach we followed, from combining logarithms to checking for extraneous solutions, is a systematic method applicable to many similar problems. This process not only provides the solution to the specific equation but also reinforces the underlying mathematical principles. Mastery of these techniques is crucial for students and professionals alike, as logarithmic equations frequently appear in various scientific and engineering contexts. By practicing and understanding these methods, one can confidently tackle complex mathematical problems and apply these skills in practical situations.