Solving 2 Log₄ X - Log₄ 9 = 2 Equivalent Equations Explained

Jul 9, 2025 by ADMIN 61 views

In this article, we will delve into solving the logarithmic equation 2 log₄ x - log₄ 9 = 2. Our primary goal is to identify which of the provided options is equivalent to the given equation. This requires a thorough understanding of logarithmic properties and how to manipulate them effectively. We will meticulously analyze the equation, applying relevant logarithmic rules to transform it into a simpler, equivalent form. By doing so, we can accurately determine the correct option among the choices provided. We aim to provide a clear, step-by-step solution that not only answers the question but also enhances your comprehension of logarithmic equations and their manipulation. Logarithmic equations play a crucial role in various fields, including mathematics, physics, and engineering, making it essential to master the techniques for solving them. This discussion will serve as a comprehensive guide to tackling similar problems, reinforcing your skills in this area of mathematics. Our approach will involve breaking down the problem into manageable steps, explaining the rationale behind each step, and highlighting common pitfalls to avoid. By the end of this article, you will have a solid grasp of how to solve logarithmic equations and be well-equipped to tackle more complex problems.

Understanding Logarithmic Properties

Before diving into the solution, it's crucial to understand the fundamental logarithmic properties that govern the manipulation of logarithmic expressions. These properties are the building blocks for solving logarithmic equations and simplifying complex expressions. The most relevant properties for this problem are the power rule and the quotient rule. The power rule states that logₐ(xⁿ) = n logₐ(x), which allows us to move exponents inside a logarithm as coefficients outside, and vice versa. This property is vital for handling the term 2 log₄ x in our equation. The quotient rule, on the other hand, states that logₐ(x) - logₐ(y) = logₐ(x/y), which enables us to combine two logarithmic terms with the same base that are being subtracted. This rule will be instrumental in combining the terms 2 log₄ x and log₄ 9. Understanding these properties thoroughly is not just about memorizing formulas; it's about grasping the underlying principles that allow us to transform logarithmic expressions effectively. Without a solid understanding of these properties, solving logarithmic equations can become a daunting task. By mastering these rules, you gain the ability to simplify complex expressions, isolate variables, and ultimately solve the equation. Moreover, these properties are not limited to solving equations; they are also essential in simplifying expressions in calculus, physics, and other advanced fields. Therefore, investing time in understanding and applying these properties is a valuable endeavor for any student of mathematics or related disciplines. In the following sections, we will demonstrate how these properties are applied step-by-step to solve the given logarithmic equation, providing a practical example of their utility.

Step-by-Step Solution

Now, let's proceed with the step-by-step solution to the equation 2 log₄ x - log₄ 9 = 2. Our first step involves applying the power rule of logarithms to the term 2 log₄ x. According to the power rule, n logₐ(x) = logₐ(xⁿ). Applying this rule, we can rewrite 2 log₄ x as log₄(x²). So, our equation now becomes log₄(x²) - log₄ 9 = 2. This transformation is crucial because it allows us to combine the logarithmic terms using the quotient rule. The next step is to apply the quotient rule, which states that logₐ(x) - logₐ(y) = logₐ(x/y). Applying this rule to our equation, we combine log₄(x²) and log₄ 9 to get log₄(x²/9) = 2. This step significantly simplifies the equation, reducing it to a single logarithmic term. Now, we need to eliminate the logarithm to solve for x. To do this, we rewrite the equation in exponential form. The equation log₄(x²/9) = 2 is equivalent to 4² = x²/9. This transformation is based on the fundamental relationship between logarithms and exponentials. The base of the logarithm becomes the base of the exponent, the result of the logarithm becomes the exponent, and the argument of the logarithm becomes the result of the exponentiation. Simplifying 4² gives us 16 = x²/9. To isolate x², we multiply both sides of the equation by 9, resulting in 144 = x². Finally, we take the square root of both sides to solve for x. This gives us x = ±12. However, we must consider the domain of the logarithmic function. The argument of a logarithm must be positive. In the original equation, we have log₄ x, which means x must be greater than 0. Therefore, we discard the negative solution, x = -12, leaving us with x = 12. This step is crucial because it ensures that our solution is valid within the context of the original equation. We have now successfully solved the equation, arriving at the solution x = 12. In the following sections, we will compare our solution with the provided options to identify the correct equivalent equation.

Identifying the Equivalent Equation

Having solved the equation 2 log₄ x - log₄ 9 = 2, our next task is to identify the equivalent equation among the given options. We transformed the original equation into log₄(x²/9) = 2. This form directly corresponds to one of the provided options, making the identification straightforward. Let's revisit the options:

A. log₄(2x/9) = 2
B. log₄(x² - 9) = 2
C. log₄(x²/9) = 2

By comparing our transformed equation, log₄(x²/9) = 2, with the options, it's clear that option C, log₄(x²/9) = 2, is the correct equivalent equation. This step underscores the importance of simplifying the original equation to a form that can be directly compared with the provided options. The ability to manipulate logarithmic expressions and identify equivalent forms is a crucial skill in solving mathematical problems. It not only helps in finding the correct answer but also enhances your understanding of the underlying mathematical concepts. Options A and B are not equivalent to the original equation. Option A, log₄(2x/9) = 2, is incorrect because it does not accurately reflect the application of the power rule and the quotient rule to the original equation. Option B, log₄(x² - 9) = 2, is also incorrect because it implies a different order of operations and a misunderstanding of logarithmic properties. The correct option, C, precisely captures the result of applying the power rule and the quotient rule, demonstrating a clear understanding of logarithmic transformations. In the next section, we will summarize our findings and highlight the key steps involved in solving the problem, reinforcing the concepts discussed and providing a comprehensive overview of the solution process.

Conclusion

In conclusion, we successfully identified the equation equivalent to 2 log₄ x - log₄ 9 = 2. Through a step-by-step application of logarithmic properties, we transformed the original equation into a simpler, equivalent form. We began by applying the power rule to rewrite 2 log₄ x as log₄(x²). This allowed us to combine the logarithmic terms using the quotient rule, resulting in log₄(x²/9) = 2. By comparing this transformed equation with the provided options, we clearly identified option C, log₄(x²/9) = 2, as the correct equivalent equation. This process highlights the importance of understanding and applying logarithmic properties effectively. The power rule and the quotient rule are fundamental tools in manipulating logarithmic expressions, and mastering their application is crucial for solving logarithmic equations. Furthermore, we emphasized the importance of considering the domain of the logarithmic function when solving for x. In this case, we discarded the negative solution because the argument of a logarithm must be positive. This step underscores the need for careful consideration of the mathematical context when interpreting solutions. Solving logarithmic equations is a valuable skill in various mathematical and scientific disciplines. It requires a combination of algebraic manipulation, understanding of logarithmic properties, and attention to detail. By breaking down the problem into manageable steps and explaining the rationale behind each step, we aimed to provide a clear and comprehensive guide to solving similar problems. The ability to transform equations, apply relevant rules, and identify equivalent forms is a fundamental aspect of mathematical problem-solving. We hope that this article has not only provided the solution to the specific problem but also enhanced your understanding of logarithmic equations and their manipulation. By mastering these concepts, you will be well-equipped to tackle more complex problems and excel in your mathematical endeavors.

In summary, the equivalent equation is:

C. log₄(x²/9) = 2