Solving 2 Log₅ X = Log₅ 4 Logarithmic Equation

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Introduction

In this comprehensive guide, we will delve into the process of solving the logarithmic equation 2 log₅ x = log₅ 4. This equation falls under the domain of mathematics, specifically focusing on logarithms and their properties. Logarithmic equations often appear in various mathematical contexts, including algebra, calculus, and applied sciences. Understanding how to solve them is crucial for anyone pursuing studies or careers in these fields. Our goal is to not only find the correct solutions but also to provide a step-by-step explanation that enhances understanding and problem-solving skills. This exploration will cover the fundamental properties of logarithms, the techniques used to simplify and solve logarithmic equations, and the importance of verifying solutions to avoid extraneous results.

This detailed explanation aims to assist students, educators, and anyone interested in mathematics in grasping the concepts involved. By the end of this guide, you will have a solid understanding of how to approach and solve similar logarithmic equations confidently. We will explore common pitfalls and errors that students often make, providing insights to help you avoid them. Additionally, we will emphasize the significance of understanding the domain of logarithmic functions, ensuring that the solutions obtained are valid within the given mathematical context. Let’s embark on this journey to master logarithmic equations and their solutions.

Understanding Logarithmic Equations

To effectively solve the logarithmic equation 2 log₅ x = log₅ 4, we must first establish a solid understanding of the fundamentals of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, the logarithm of a number x with respect to a base b is the exponent to which b must be raised to produce x. This is mathematically represented as logb(x) = y, which is equivalent to by = x. Here, b is the base of the logarithm, and x is the argument. The base b must be a positive number not equal to 1, and the argument x must be positive, as logarithms are not defined for non-positive numbers.

Logarithmic equations often require manipulation using various logarithmic properties to simplify and isolate the variable. These properties include the power rule, the product rule, and the quotient rule. The power rule states that logb(xn) = n logb(x), which is particularly relevant to our equation. The product rule states that logb(mn) = logb(m) + logb(n), and the quotient rule states that logb(m/n) = logb(m) - logb(n). Understanding and applying these properties correctly is crucial for simplifying complex logarithmic expressions. Additionally, the change of base formula, logb(a) = logc(a) / logc(b), is a valuable tool when dealing with logarithms of different bases. By mastering these basic principles and properties, you will be well-equipped to tackle a wide range of logarithmic equations and problems. This foundation is essential for understanding the step-by-step solution we will discuss in the subsequent sections.

Step-by-Step Solution

Now, let's solve the given logarithmic equation 2 log₅ x = log₅ 4 step by step. This process will involve applying the properties of logarithms to simplify the equation and isolate the variable x. We will meticulously go through each step, explaining the rationale behind the actions taken.

  1. Apply the Power Rule of Logarithms: The power rule states that logb(xn) = n logb(x). We can use this rule to simplify the left side of the equation. The term 2 log₅ x can be rewritten as log₅ (x²). Applying this rule transforms the equation into: log₅ (x²) = log₅ 4

  2. Equate the Arguments: Since the logarithms on both sides of the equation have the same base (base 5), we can equate the arguments. This is based on the property that if logb(m) = logb(n), then m = n. Thus, we set the expressions inside the logarithms equal to each other: x² = 4

  3. Solve the Quadratic Equation: We now have a simple quadratic equation. To solve for x, we take the square root of both sides of the equation: √(x²) = ±√4 x = ±2

  4. Check for Extraneous Solutions: Logarithms are only defined for positive arguments. This means that x must be greater than zero in the original equation, 2 log₅ x = log₅ 4. Therefore, we need to check both solutions, x = 2 and x = -2, to ensure they are valid.

    • For x = 2: 2 log₅ (2) = log₅ (2²) = log₅ (4). This solution is valid.
    • For x = -2: Logarithms are not defined for negative numbers. Therefore, x = -2 is an extraneous solution and must be discarded.
  5. Final Solution: After checking for extraneous solutions, we find that the only valid solution is x = 2. Thus, the solution to the equation 2 log₅ x = log₅ 4 is x = 2.

By following these steps, we have successfully solved the logarithmic equation. This detailed explanation underscores the importance of understanding logarithmic properties and the necessity of checking solutions to avoid extraneous results. The correct application of these principles ensures accurate and valid solutions.

Checking for Extraneous Solutions

Checking for extraneous solutions is a crucial step in solving logarithmic equations. Extraneous solutions are solutions that emerge during the solving process but do not satisfy the original equation. These can occur because logarithmic functions have restrictions on their domains. Specifically, the argument of a logarithm must be positive. Therefore, any solution that results in a non-positive argument in the original equation is extraneous.

In the equation 2 log₅ x = log₅ 4, we found two potential solutions: x = 2 and x = -2. To verify these, we need to substitute each value back into the original equation and check if the equation holds true. Additionally, we must ensure that the argument of the logarithm remains positive. For x = 2, the argument in the original equation is simply x, which is 2, a positive number. Substituting x = 2 into the original equation gives:

2 log₅ (2) = log₅ (2²) = log₅ (4)

This confirms that x = 2 is a valid solution.

However, for x = -2, the argument in the original equation is -2. Since the logarithm of a negative number is undefined, x = -2 cannot be a valid solution. This is an example of an extraneous solution. It is important to recognize and eliminate such solutions to ensure the accuracy of the final result. The process of checking for extraneous solutions highlights the need for a thorough understanding of the properties and limitations of logarithmic functions. It is a common mistake to overlook this step, which can lead to incorrect answers. Therefore, always verify the solutions by substituting them back into the original equation and checking the domain restrictions.

Common Mistakes and How to Avoid Them

When solving logarithmic equations, several common mistakes can lead to incorrect solutions. Recognizing these pitfalls and understanding how to avoid them is crucial for mastering logarithmic problem-solving. One of the most frequent errors is forgetting to check for extraneous solutions. As previously discussed, extraneous solutions arise when a potential solution violates the domain restrictions of the logarithmic function, specifically the requirement that the argument of the logarithm must be positive. Always substitute the solutions back into the original equation to confirm their validity.

Another common mistake is the incorrect application of logarithmic properties. For instance, students may confuse the power rule, product rule, and quotient rule, leading to incorrect simplifications. It is essential to have a firm grasp of these rules and apply them precisely. A good practice is to write down the logarithmic properties before solving the equation to ensure they are correctly applied. For example, in the equation 2 log₅ x = log₅ 4, if the power rule is misapplied, it can lead to an entirely different solution. Similarly, errors can occur when dealing with logarithms of different bases. Remember to use the change of base formula when necessary and ensure that all logarithms are expressed in the same base before proceeding with the solution.

Another pitfall is misinterpreting the order of operations. For example, when simplifying expressions involving logarithms, ensure that logarithmic operations are performed before arithmetic operations. Ignoring this order can lead to incorrect results. Finally, making algebraic errors while manipulating the equation is a common mistake. This includes errors in arithmetic, factoring, and simplification. To avoid such errors, it is beneficial to work methodically, showing each step clearly and double-checking the calculations. By being mindful of these common mistakes and adopting careful problem-solving practices, you can improve your accuracy and confidence in solving logarithmic equations.

Conclusion

In conclusion, solving logarithmic equations requires a solid understanding of logarithmic properties, careful application of these properties, and diligence in checking for extraneous solutions. Throughout this guide, we have meticulously addressed the solution of the equation 2 log₅ x = log₅ 4. We began by establishing the fundamental principles of logarithms, emphasizing the inverse relationship between logarithms and exponentiation and the crucial properties that govern logarithmic operations. These include the power rule, the product rule, and the quotient rule, each playing a vital role in simplifying and manipulating logarithmic expressions.

We then provided a detailed step-by-step solution to the given equation. This involved applying the power rule to condense the logarithmic term, equating the arguments of the logarithms due to their shared base, solving the resulting quadratic equation, and, most importantly, checking for extraneous solutions. This step-by-step approach underscored the significance of each action and the underlying mathematical reasoning. We highlighted the necessity of checking solutions against the domain of logarithmic functions, which mandates that the argument of a logarithm must be positive. This process helped us identify and discard the extraneous solution x = -2, confirming x = 2 as the sole valid solution.

Furthermore, we addressed common mistakes that students often make when solving logarithmic equations. These include overlooking the checking of extraneous solutions, misapplying logarithmic properties, and making algebraic errors. By being aware of these pitfalls and adopting careful problem-solving practices, learners can enhance their accuracy and confidence in tackling logarithmic problems. The ultimate goal is to develop a comprehensive understanding of logarithms that extends beyond rote memorization, fostering the ability to approach and solve a wide range of logarithmic equations effectively. By mastering these skills, you can confidently apply them in various mathematical and real-world contexts.