Solving Logarithmic Equations A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into a fascinating logarithmic equation: 3 ln 2 + ln 8 = 2 ln(4x). This equation might seem intimidating at first glance, but fear not! We're going to break it down step by step, ensuring you not only understand the solution but also grasp the underlying principles of logarithmic operations. So, buckle up, grab your thinking caps, and let's embark on this mathematical journey together!

Understanding the Fundamentals of Logarithms

Before we jump into solving the equation, it's crucial to have a solid grasp of the fundamentals of logarithms. Logarithms, in essence, are the inverse operations of exponentiation. Think of it this way: if 2 raised to the power of 3 equals 8 (2³ = 8), then the logarithm base 2 of 8 is 3 (log₂ 8 = 3). In simpler terms, a logarithm answers the question: "What exponent do I need to raise the base to in order to get this number?" When we see "ln", it signifies the natural logarithm, which has a base of e (Euler's number, approximately 2.71828). Natural logarithms are widely used in calculus, physics, and various other scientific fields, making them a crucial concept to master.

The core properties of logarithms are our secret weapons in solving equations like this one. Let's highlight a few key properties that we'll be using today:

• Product Rule: ln(a * b) = ln(a) + ln(b)
• Quotient Rule: ln(a / b) = ln(a) - ln(b)
• Power Rule: ln(a^b) = b * ln(a)

These rules allow us to manipulate logarithmic expressions, simplifying them and making them easier to work with. The power rule, in particular, will be instrumental in tackling the equation at hand. By understanding these logarithmic properties, you're not just memorizing formulas; you're unlocking a powerful toolkit for solving a wide array of mathematical problems. Remember, guys, the beauty of math lies in understanding the "why" behind the "how". So, let's keep these fundamentals in mind as we move forward.

Cracking the Code: Solving 3 ln 2 + ln 8 = 2 ln(4x)

Now, let's get our hands dirty and solve the equation 3 ln 2 + ln 8 = 2 ln(4x). Remember, the goal here isn't just to arrive at the correct answer, but also to understand the process. We'll walk through each step meticulously, explaining the reasoning behind every move. First things first, we need to simplify both sides of the equation using those handy logarithmic properties we just discussed.

Let's tackle the left-hand side (LHS) first: 3 ln 2 + ln 8. Notice that we have a constant multiplying a logarithm (3 ln 2). This is where the power rule comes into play. We can rewrite 3 ln 2 as ln(2³), which simplifies to ln 8. So, our LHS now becomes ln 8 + ln 8. This looks much simpler already, doesn't it? Now, we can use the product rule to combine these two logarithmic terms. Remember, ln(a) + ln(b) = ln(a * b). Applying this rule, we get ln(8 * 8), which simplifies to ln 64. So, the left-hand side of our equation has been beautifully transformed into ln 64.

Now, let's shift our focus to the right-hand side (RHS): 2 ln(4x). Again, we see a constant multiplying a logarithm, so we'll use the power rule. We can rewrite 2 ln(4x) as ln((4x)²). Now, we need to simplify the expression inside the logarithm. (4x)² is simply 4x multiplied by itself, which equals 16x². So, the RHS now looks like ln(16x²). Great! We've successfully simplified both sides of the equation using our logarithmic superpowers.

Our equation now stands as ln 64 = ln(16x²). This is a pivotal moment. Since we have the natural logarithm on both sides of the equation, we can confidently say that the arguments inside the logarithms must be equal. In other words, if ln(a) = ln(b), then a = b. Applying this principle, we can equate the arguments: 64 = 16x². We've effectively eliminated the logarithms and are left with a simple algebraic equation. This is where our basic algebra skills come into play. To solve for x, we first divide both sides of the equation by 16. This gives us 4 = x². Now, to isolate x, we take the square root of both sides. Remember, when taking the square root, we need to consider both the positive and negative solutions. So, we get x = ±2.

But hold on a second! We're not quite done yet. We need to check our solutions to make sure they are valid within the context of the original equation. Remember, the argument of a logarithm must be positive. Let's plug our solutions back into the original equation and see if they hold up.

The Moment of Truth: Verifying the Solution

We've arrived at two potential solutions for our equation: x = 2 and x = -2. Now comes the crucial step: verifying these solutions. Remember that in the realm of logarithms, the argument (the value inside the logarithm) must always be positive. This is a critical constraint that we need to keep in mind. Let's start by plugging x = 2 back into the original equation: 3 ln 2 + ln 8 = 2 ln(4x). Substituting x = 2, we get 3 ln 2 + ln 8 = 2 ln(4 * 2), which simplifies to 3 ln 2 + ln 8 = 2 ln 8. Let's further simplify the left-hand side. We know that 3 ln 2 is equal to ln(2³) or ln 8. So, the equation becomes ln 8 + ln 8 = 2 ln 8. We can combine the terms on the left using the product rule: ln(8 * 8) = 2 ln 8, which simplifies to ln 64 = 2 ln 8. Using the power rule on the right side, we get ln 64 = ln(8²), which further simplifies to ln 64 = ln 64. Voila! The equation holds true for x = 2. This means x = 2 is indeed a valid solution.

Now, let's put x = -2 to the test. Plugging it into the original equation, we get 3 ln 2 + ln 8 = 2 ln(4 * -2), which simplifies to 3 ln 2 + ln 8 = 2 ln(-8). Uh oh! We've encountered a problem. We have a logarithm of a negative number (ln(-8)). As we discussed earlier, the argument of a logarithm must be positive. Therefore, ln(-8) is undefined in the real number system. This immediately disqualifies x = -2 as a solution. It's crucial to remember this constraint when dealing with logarithmic equations. Always check your solutions to ensure they don't lead to taking the logarithm of a non-positive number.

So, after rigorous verification, we've concluded that only one solution holds true: x = 2. The negative solution, x = -2, is an extraneous solution that arises from the algebraic manipulation but doesn't satisfy the original logarithmic equation. This highlights the importance of verification in problem-solving. It's not enough to just arrive at an answer; we need to ensure it's a valid answer within the given context.

The True Solution and Key Takeaways

After our detailed exploration, we've definitively determined that the true solution to the equation 3 ln 2 + ln 8 = 2 ln(4x) is x = 2. We embarked on this journey by revisiting the fundamental properties of logarithms, which allowed us to simplify the equation strategically. We then applied the power rule and the product rule to condense logarithmic expressions, ultimately leading us to a simple algebraic equation. Solving this equation gave us two potential solutions, but the crucial step of verification revealed that only x = 2 is a valid solution.

The extraneous solution, x = -2, served as a valuable reminder of the importance of checking our work, especially when dealing with logarithms. The constraint that the argument of a logarithm must be positive is a non-negotiable rule. Ignoring it can lead to incorrect conclusions and a false sense of accomplishment. So, always remember to plug your solutions back into the original equation and make sure they make sense within the context of the problem.

This problem beautifully illustrates the power and elegance of logarithms. By understanding the properties and applying them judiciously, we can transform seemingly complex equations into manageable forms. But more than just solving a specific equation, we've reinforced the importance of a methodical approach to problem-solving. We've seen how a strong foundation in fundamental concepts, combined with careful execution and rigorous verification, can lead us to the correct answer every time. So, keep practicing, keep exploring, and keep those logarithmic skills sharp!