Solving Logarithmic Equations 2 Log₃(x) = 4

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Introduction

In this article, we will delve into the intricacies of solving the logarithmic equation 2 log₃(x) = 4. Logarithmic equations, while sometimes appearing complex, are fundamental in mathematics and have applications in various fields such as physics, engineering, and computer science. We will explore the step-by-step process of solving this particular equation, analyze the given options, and discuss the concepts of extraneous solutions. Our goal is to provide a clear and comprehensive understanding of the solution, ensuring that readers can confidently tackle similar problems in the future. Understanding logarithmic equations is not just about finding the correct answer; it's about grasping the underlying principles that govern these mathematical expressions. This article aims to enhance your problem-solving skills and deepen your appreciation for the elegance of logarithmic functions.

Understanding Logarithmic Equations

Before diving into the specific equation 2 log₃(x) = 4, it's crucial to have a solid grasp of what logarithmic equations are and the rules that govern them. A logarithmic equation is an equation that involves a logarithm of an expression. The logarithm is the inverse operation to exponentiation. The most common types of logarithms are the common logarithm (base 10) and the natural logarithm (base e), but logarithms can have any positive base not equal to 1. The equation we are dealing with, 2 log₃(x) = 4, involves a logarithm with base 3. The logarithmic function log₃(x) asks the question: "To what power must we raise 3 to get x?" Understanding this fundamental question is key to unlocking the solution. Logarithmic equations often require careful manipulation to isolate the variable, and it's essential to be mindful of the domain of logarithmic functions, as logarithms are only defined for positive arguments. This means that when we solve for x, we must check that our solutions do not result in taking the logarithm of a non-positive number. In the next section, we will begin the process of solving the equation, applying the properties of logarithms to simplify it and isolate the variable x.

Step-by-Step Solution of 2 log₃(x) = 4

Let's embark on the journey of solving the logarithmic equation 2 log₃(x) = 4 step by step. This meticulous approach will not only lead us to the correct solution but also reinforce the fundamental principles of logarithmic equations.

  1. Initial Equation: The equation we start with is 2 log₃(x) = 4. Our primary goal is to isolate the logarithmic term.
  2. Divide by the Coefficient: To isolate the logarithmic term, we divide both sides of the equation by 2. This gives us log₃(x) = 2.
  3. Convert to Exponential Form: Now, we convert the logarithmic equation to its equivalent exponential form. The logarithmic equation log₃(x) = 2 is equivalent to the exponential equation 3² = x. This step is crucial as it transforms the equation into a more manageable form.
  4. Solve for x: Evaluating 3² gives us x = 9. This is our potential solution. However, it's important to remember that we need to check this solution in the original equation to ensure it is valid and not an extraneous solution.
  5. Verification: Substitute x = 9 back into the original equation: 2 log₃(9) = 4. Since 9 is 3², log₃(9) = 2, and thus 2 * 2 = 4, which confirms that x = 9 is indeed a valid solution.

This methodical approach ensures that we not only find the solution but also understand the reasoning behind each step. In the next section, we will examine the given options and determine which ones are true based on our solution process.

Analyzing the Options

Now that we have solved the equation 2 log₃(x) = 4 and found the solution x = 9, let's analyze the provided options to determine which statements are true. This exercise will not only solidify our understanding of the solution but also enhance our ability to interpret and evaluate mathematical statements.

  • A) log₃(x²) = 4: This option can be derived from the original equation using the power rule of logarithms. The power rule states that logₐ(bᶜ) = c * logₐ(b). Applying this rule to our original equation, 2 log₃(x) = 4, we can rewrite it as log₃(x²) = 4. Thus, this statement is true.
  • B) x² = 3⁴: This option is a direct consequence of option A. If log₃(x²) = 4, then converting this logarithmic equation to exponential form gives us x² = 3⁴. Hence, this statement is also true.
  • C) x² = 81: This option is simply the evaluation of 3⁴ in option B. Since 3⁴ = 81, x² = 81 is a true statement.
  • D) x = 9, -9: This option lists two potential solutions. While x = 9 is the solution we found, x = -9 needs to be carefully considered. In the context of the original logarithmic equation, 2 log₃(x) = 4, substituting x = -9 results in taking the logarithm of a negative number, which is undefined. Therefore, x = -9 is not a valid solution.
  • E) x = 9 and x = -9 are true solutions: Based on our analysis, this statement is false because x = -9 is not a valid solution.
  • F) x = 9 and x = -9 are extraneous solutions: This statement is partially correct. While x = -9 is an extraneous solution, x = 9 is a valid solution. Thus, the statement is not entirely accurate.
  • G) x = 9 is: This statement is incomplete but implies that x = 9 is a true solution, which we have verified.

Through this analysis, we have not only identified the true statements but also reinforced the importance of checking for extraneous solutions in logarithmic equations. In the next section, we will delve deeper into the concept of extraneous solutions and why they arise in the context of logarithmic equations.

Extraneous Solutions in Logarithmic Equations

The concept of extraneous solutions is particularly important when dealing with logarithmic and radical equations. An extraneous solution is a value that satisfies the transformed equation but not the original equation. In the context of logarithmic equations, extraneous solutions often arise due to the domain restrictions of logarithmic functions. Logarithms are only defined for positive arguments; that is, we can only take the logarithm of a positive number. When we manipulate logarithmic equations, we sometimes introduce solutions that, when plugged back into the original equation, result in taking the logarithm of a negative number or zero, which is undefined.

In our example, 2 log₃(x) = 4, we found a potential solution of x = 9. However, if we had followed a slightly different path in solving the equation, such as squaring both sides at some point, we might have obtained x = -9 as another potential solution. Substituting x = -9 into the original equation gives us 2 log₃(-9), which is undefined because we cannot take the logarithm of a negative number. Thus, x = -9 is an extraneous solution.

The key takeaway here is that whenever we solve logarithmic equations, it is crucial to check all potential solutions in the original equation. This step ensures that we discard any extraneous solutions and only accept the values that satisfy the original logarithmic equation. Understanding extraneous solutions is not just a matter of mathematical correctness; it reflects a deeper understanding of the nature and limitations of logarithmic functions. In the concluding section, we will summarize our findings and reiterate the key points in solving logarithmic equations.

Conclusion

In summary, solving the logarithmic equation 2 log₃(x) = 4 has been a comprehensive exercise that highlights several important aspects of logarithmic equations. We began by understanding the fundamental definition of logarithms and their properties. We then meticulously solved the equation step by step, arriving at the solution x = 9. Crucially, we analyzed the given options, determining which statements were true based on our solution. This analysis reinforced the importance of careful evaluation and interpretation of mathematical statements.

A significant part of our discussion was dedicated to the concept of extraneous solutions. We learned that extraneous solutions can arise in logarithmic equations due to the domain restrictions of logarithmic functions. The act of checking potential solutions in the original equation is therefore not just a formality but a necessary step to ensure the validity of our results. This process underscores the importance of understanding the underlying principles of logarithmic functions and their limitations.

Ultimately, the ability to solve logarithmic equations is a valuable skill in mathematics and its applications. It requires a combination of algebraic manipulation, a solid understanding of logarithmic properties, and a keen awareness of potential pitfalls such as extraneous solutions. By mastering these skills, one can confidently approach a wide range of mathematical problems and appreciate the elegance and power of logarithmic functions.