Solving 3ln2 + Ln8 = 2ln(4x) A Step-by-Step Solution

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Finding solutions to logarithmic equations might seem daunting at first, but with a clear understanding of logarithmic properties and a systematic approach, these problems become quite manageable. In this article, we will delve deep into the equation 3ln2 + ln8 = 2ln(4x), dissecting each step to arrive at the true solution. We will explore the underlying logarithmic principles, apply them methodically, and clarify any potential areas of confusion. Our goal is not just to provide the answer but to equip you with the knowledge and skills to tackle similar logarithmic challenges with confidence.

Decoding the Logarithmic Equation

Before we jump into solving the equation, let's first understand the fundamental logarithmic properties that will guide our journey. Logarithms, in essence, are the inverse operations of exponentiation. The expression ln(x) represents the natural logarithm of x, which is the logarithm to the base e (Euler's number, approximately 2.71828). Understanding this basic definition is crucial for manipulating logarithmic expressions effectively. Several key properties of logarithms will be instrumental in solving our equation:

  1. Power Rule: ln(ab) = bln(a)*
  2. Product Rule: ln(a * b) = ln(a) + ln(b)
  3. Quotient Rule: ln(a / b) = ln(a) - ln(b)
  4. Logarithmic Equality: If ln(a) = ln(b), then a = b

With these properties in our arsenal, we are well-prepared to tackle the equation at hand. The given equation, 3ln2 + ln8 = 2ln(4x), involves natural logarithms and requires careful application of these properties to simplify and isolate the variable x. Our step-by-step approach will ensure clarity and accuracy in finding the true solution. We'll begin by simplifying each side of the equation independently, utilizing the power rule to handle coefficients and the product rule to combine logarithmic terms. This methodical approach will lead us to a point where we can confidently apply the logarithmic equality property, ultimately revealing the value of x that satisfies the equation.

Step-by-Step Solution to 3ln2 + ln8 = 2ln(4x)

Now, let's embark on the journey of solving the equation 3ln2 + ln8 = 2ln(4x). We'll break down the process into manageable steps, ensuring that each manipulation is clearly explained and justified. This meticulous approach will not only lead us to the correct answer but also solidify your understanding of logarithmic problem-solving techniques.

Step 1: Applying the Power Rule

The first step involves utilizing the power rule of logarithms, which states that ln(ab) = bln(a)*. We can apply this rule to the terms with coefficients in front of the logarithms. Specifically, we have 3ln2 and 2ln(4x). Applying the power rule, we transform these terms as follows:

  • 3ln2 becomes ln(23), which simplifies to ln8
  • 2ln(4x) becomes ln((4x)2), which simplifies to ln(16x2)

Substituting these simplified terms back into the original equation, we now have:

ln8 + ln8 = ln(16x2)

This transformation is a crucial step in consolidating the logarithmic terms, making the equation easier to manage. By eliminating the coefficients in front of the logarithms, we pave the way for the next step, where we will employ the product rule to further simplify the left side of the equation.

Step 2: Employing the Product Rule

The next strategic move is to leverage the product rule of logarithms. This rule, as we've discussed, states that ln(a * b) = ln(a) + ln(b). On the left side of our equation, we have the sum of two logarithmic terms: ln8 + ln8. Applying the product rule, we can combine these terms into a single logarithm:

ln8 + ln8 = ln(8 * 8) = ln64

Now, our equation looks significantly simpler:

ln64 = ln(16x2)

The product rule has effectively reduced the complexity of the equation, bringing us closer to isolating the variable x. By combining the logarithmic terms on the left side, we've created a balanced equation where we can directly compare the arguments of the logarithms. This simplification sets the stage for the crucial step of applying the logarithmic equality property.

Step 3: Utilizing Logarithmic Equality

The logarithmic equality property is a cornerstone in solving logarithmic equations. It states that if ln(a) = ln(b), then a = b. In our transformed equation, ln64 = ln(16x2), we have logarithms with the same base (natural logarithm) on both sides. Therefore, we can confidently equate the arguments:

64 = 16x2

This step is pivotal as it eliminates the logarithms altogether, transforming the equation into a simple algebraic equation. By equating the arguments, we've created a direct relationship between the constant 64 and the expression involving x. This algebraic equation is now easily solvable using basic algebraic techniques. We've successfully navigated the logarithmic complexities and arrived at a point where we can directly solve for x.

Step 4: Solving for x

With the equation simplified to 64 = 16x2, we can now proceed to isolate x and find its value. This involves a few basic algebraic steps that will lead us to the solution.

First, we divide both sides of the equation by 16:

64 / 16 = 16x2 / 16

This simplifies to:

4 = x2

Next, we take the square root of both sides of the equation. Remember that taking the square root can result in both positive and negative solutions:

√4 = √(x2)

This gives us two possible solutions:

x = ±2

However, we must be cautious and consider the domain of the original logarithmic equation. Logarithms are only defined for positive arguments. In the original equation, we have ln(4x). If x = -2, then 4x = -8, and the logarithm of a negative number is undefined. Therefore, we must discard the negative solution.

Step 5: Verifying the Solution

Therefore, the only valid solution is x = 2. It's always good practice to verify the solution by plugging it back into the original equation to ensure it holds true. Substituting x = 2 into the original equation:

3ln2 + ln8 = 2ln(4 * 2)

3ln2 + ln8 = 2ln8

We know that ln8 = ln(23) = 3ln2, so we can rewrite the left side as:

3ln2 + 3ln2 = 2(3ln2)

6ln2 = 6ln2

This confirms that our solution, x = 2, is indeed correct. We have successfully navigated the logarithmic equation, applied the necessary properties, and arrived at the true solution.

The True Solution: x = 2

After meticulously applying logarithmic properties and solving the resulting algebraic equation, we have arrived at the solution to the equation 3ln2 + ln8 = 2ln(4x). The true solution is x = 2. This solution not only satisfies the equation but also respects the domain restrictions inherent in logarithmic functions. Our journey through this problem has highlighted the importance of understanding logarithmic properties, applying them systematically, and verifying the solution to ensure its validity.

Mastering Logarithmic Equations: Tips and Tricks

Solving logarithmic equations can become second nature with practice and a solid understanding of the underlying principles. Here are some valuable tips and tricks to help you master these types of problems:

  1. Know Your Properties: The logarithmic properties (power rule, product rule, quotient rule, and equality property) are your most powerful tools. Memorize them and understand how to apply them in different situations.
  2. Simplify First: Before attempting to isolate the variable, simplify the equation as much as possible. Use logarithmic properties to combine terms, eliminate coefficients, and reduce complexity.
  3. Isolate the Logarithm: If possible, isolate the logarithmic term on one side of the equation. This makes it easier to apply the exponential function (or the logarithmic equality property) to solve for the variable.
  4. Check for Extraneous Solutions: Always remember to check your solutions in the original equation. Logarithmic equations can sometimes produce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. This is particularly important when dealing with expressions that could result in logarithms of negative numbers or zero.
  5. Practice Regularly: The key to mastering any mathematical concept is consistent practice. Work through a variety of logarithmic equations, gradually increasing the difficulty. This will help you develop your problem-solving skills and build confidence.

By incorporating these tips and tricks into your problem-solving approach, you'll be well-equipped to tackle a wide range of logarithmic challenges. Remember, the journey to mastery involves understanding the fundamentals, applying them consistently, and learning from your experiences.

Conclusion: Embracing the Power of Logarithms

In this comprehensive exploration, we have successfully navigated the intricacies of the logarithmic equation 3ln2 + ln8 = 2ln(4x), ultimately revealing the true solution: x = 2. Our journey has underscored the importance of a solid grasp of logarithmic properties, a systematic approach to problem-solving, and the crucial step of verifying solutions. Logarithms, while sometimes perceived as complex, are powerful tools in mathematics and various scientific disciplines. By mastering logarithmic equations, you unlock a deeper understanding of mathematical relationships and enhance your problem-solving capabilities.

Remember, the key to success in mathematics lies not just in finding the right answer but in understanding the process behind it. Embrace the challenge, practice consistently, and celebrate the joy of unraveling mathematical mysteries. With dedication and the right approach, you can conquer any logarithmic equation and confidently apply these principles to a broader range of mathematical endeavors. The world of logarithms awaits your exploration – delve in and discover its power!