Solving The Logarithmic Equation 3 Ln 2 + Ln 8 = 2 Ln (4x)

by ADMIN 59 views

Introduction: Delving into Logarithmic Equations

In the realm of mathematics, logarithmic equations often present themselves as intriguing puzzles, demanding a keen understanding of logarithmic properties and algebraic manipulation to unravel their solutions. The equation at hand, 3 ln 2 + ln 8 = 2 ln (4x), exemplifies this challenge, requiring us to navigate the intricacies of logarithms to determine the true value of x. In this comprehensive exploration, we will embark on a step-by-step journey to dissect this equation, employing the fundamental principles of logarithms to arrive at the correct solution. We will not only solve the equation but also delve into the underlying concepts and properties that govern logarithmic operations, providing a thorough understanding of the mathematical processes involved. Our aim is to equip you with the knowledge and skills to confidently tackle similar logarithmic problems in the future. This journey will involve utilizing properties such as the power rule of logarithms, the product rule, and the understanding of how to simplify logarithmic expressions. We will also emphasize the importance of checking solutions to ensure they are valid within the domain of the logarithmic functions involved. This thorough approach will ensure that you not only understand the solution to this specific problem but also gain a deeper appreciation for the elegance and power of logarithmic equations in mathematics. By the end of this article, you will have a clear grasp of how to approach and solve logarithmic equations systematically, making you more confident in your mathematical abilities.

Understanding Logarithmic Properties: The Foundation of Our Solution

Before we embark on the journey of solving the equation 3 ln 2 + ln 8 = 2 ln (4x), it's crucial to solidify our understanding of the fundamental logarithmic properties. These properties serve as the bedrock upon which our solution will be built. The natural logarithm, denoted as "ln", is a logarithm with the base e, where e is an irrational number approximately equal to 2.71828. Logarithms, in essence, are the inverse operations of exponentiation. The logarithmic properties we'll be leveraging include the power rule, the product rule, and the quotient rule. The power rule states that ln(a^b) = b ln(a), allowing us to move exponents within logarithms. The product rule tells us that ln(ab) = ln(a) + ln(b), enabling us to combine the logarithms of products. Furthermore, while not directly used in this problem, the quotient rule, ln(a/b) = ln(a) - ln(b), is an essential part of the logarithmic toolkit. To effectively apply these rules, we need to be able to identify the structure of the logarithmic expression and determine how these properties can be used to simplify it. For example, recognizing that 8 can be expressed as 2^3 is a key step in utilizing the power rule. Similarly, understanding that 4x is a product allows us to apply the product rule if necessary. Mastering these properties is not just about memorizing formulas; it's about understanding their underlying logic and how they can be strategically employed to transform and simplify logarithmic equations. This understanding will allow us to approach complex problems with a clear plan and execute the necessary steps with confidence. As we delve into the solution, we will see these properties in action, further reinforcing their importance in the context of solving logarithmic equations.

Step-by-Step Solution: Unraveling the Equation 3 ln 2 + ln 8 = 2 ln (4x)

Now, let's embark on the journey of solving the equation 3 ln 2 + ln 8 = 2 ln (4x). Our strategy will involve simplifying both sides of the equation using the logarithmic properties we've discussed and then isolating x.

  1. Simplifying the Left-Hand Side (LHS): Begin by focusing on the left side of the equation, which is 3 ln 2 + ln 8. Notice that 8 can be expressed as 2^3. Therefore, we can rewrite ln 8 as ln(2^3). Applying the power rule of logarithms, which states that ln(a^b) = b ln(a), we can further simplify ln(2^3) to 3 ln 2. Now the left side becomes 3 ln 2 + 3 ln 2. Combining these like terms, we get 6 ln 2. This simplification is a crucial first step, as it consolidates the logarithmic terms on the left side, making the equation more manageable.
  2. Simplifying the Right-Hand Side (RHS): Now, let's turn our attention to the right side of the equation, 2 ln (4x). Applying the power rule in reverse, we can rewrite 2 ln (4x) as ln ((4x)^2). Expanding (4x)^2, we get 16x^2. So, the right side of the equation becomes ln(16x^2). This transformation is essential as it removes the coefficient from the logarithm, allowing us to directly compare the arguments of the logarithmic functions on both sides of the equation.
  3. Equating and Solving: After simplifying both sides, our equation now looks like this: 6 ln 2 = ln(16x^2). Recognizing that 6 ln 2 can be written as ln(2^6) using the power rule, we have ln(2^6) = ln(16x^2). Since the logarithms are equal, their arguments must be equal as well. This gives us the equation 2^6 = 16x^2. We know that 2^6 is 64, so the equation becomes 64 = 16x^2. Dividing both sides by 16, we get x^2 = 4. Taking the square root of both sides, we find that x = ±2. However, we must consider the domain of the original logarithmic equation. Since ln(4x) appears in the original equation, 4x must be greater than 0, which means x must be greater than 0. Therefore, x = -2 is not a valid solution. The only valid solution is x = 2.

Verification: Ensuring the Solution's Validity

After arriving at a solution, it's crucial to verify its validity within the context of the original equation. This step is particularly important when dealing with logarithmic equations due to the domain restrictions of logarithmic functions. The argument of a logarithm must be strictly greater than zero. In our equation, 3 ln 2 + ln 8 = 2 ln (4x), we found that x = 2. Let's substitute this value back into the original equation to see if it holds true.

Substituting x = 2, the equation becomes:

3 ln 2 + ln 8 = 2 ln (4 * 2)

Now, we simplify both sides:

  • Left-Hand Side (LHS): 3 ln 2 + ln 8 can be rewritten as 3 ln 2 + ln (2^3). Applying the power rule, ln (2^3) becomes 3 ln 2. So, the LHS is 3 ln 2 + 3 ln 2 = 6 ln 2.
  • Right-Hand Side (RHS): 2 ln (4 * 2) simplifies to 2 ln 8. Rewriting 8 as 2^3, we have 2 ln (2^3). Applying the power rule, this becomes 2 * 3 ln 2 = 6 ln 2.

Since the left-hand side (6 ln 2) equals the right-hand side (6 ln 2), our solution x = 2 satisfies the equation. Moreover, we need to ensure that the argument of the logarithm in the original equation, 4x, remains positive when we substitute x = 2. In this case, 4 * 2 = 8, which is indeed greater than zero. Therefore, our solution x = 2 is valid. This verification step underscores the importance of not only solving the equation but also confirming that the solution aligns with the inherent constraints of logarithmic functions. By diligently verifying our solutions, we can avoid extraneous results and ensure the accuracy of our mathematical endeavors.

Conclusion: The True Solution and Key Takeaways

In conclusion, after a thorough exploration of the equation 3 ln 2 + ln 8 = 2 ln (4x), we have successfully unveiled the true solution. By meticulously applying the properties of logarithms, including the power rule and the understanding of logarithmic arguments, we arrived at the solution x = 2. We then rigorously verified this solution by substituting it back into the original equation, confirming its validity within the domain of logarithmic functions.

The correct answer is B. x = 2.

This exercise has not only provided us with a specific answer but also illuminated several key takeaways for tackling logarithmic equations:

  • Mastering Logarithmic Properties: A strong grasp of logarithmic properties, such as the power rule, product rule, and quotient rule, is paramount. These properties are the tools that enable us to manipulate and simplify logarithmic expressions.
  • Step-by-Step Simplification: Approaching the equation systematically, simplifying each side individually before attempting to solve, makes the process more manageable and reduces the likelihood of errors.
  • Verification is Crucial: Always verify your solution by substituting it back into the original equation. This step is essential to ensure that the solution is valid and that it satisfies the domain restrictions of logarithmic functions.
  • Domain Awareness: Be mindful of the domain of logarithmic functions. The argument of a logarithm must be greater than zero. This constraint can eliminate potential solutions that might arise during the algebraic manipulation process.

By internalizing these key takeaways, you will be well-equipped to confidently navigate the world of logarithmic equations, tackling challenges with clarity and precision. The journey through this equation has not only provided a solution but also a valuable learning experience, reinforcing the importance of understanding the fundamental principles of mathematics and applying them with diligence and care. As you continue your mathematical journey, remember that practice and a solid understanding of core concepts are the keys to success.