Solving 4ln(x) - 8 = 12: A Step-by-Step Guide

Have you ever stumbled upon an equation that looks a bit intimidating at first glance? Equations involving logarithms can sometimes seem tricky, but with a step-by-step approach, they become much more manageable. In this guide, we'll break down the process of solving the equation 4ln(x) - 8 = 12, exploring each step in detail so you can confidently tackle similar problems in the future. This comprehensive guide provides a clear, step-by-step solution to this equation, perfect for students and anyone looking to brush up on their algebra skills. Let's dive in!

1. Isolating the Logarithmic Term

The first crucial step in solving any equation involving logarithms is to isolate the logarithmic term. This means getting the term with the natural logarithm (ln(x) in this case) all by itself on one side of the equation. To do this, we need to get rid of the constant term (-8) that's currently cluttering things up on the left side. We achieve this by performing the inverse operation, which is adding 8 to both sides of the equation. Remember, whatever you do to one side of an equation, you must do to the other to maintain the balance and equality. When we add 8 to both sides of our equation, 4ln(x) - 8 = 12, we get a simplified equation that looks like this: 4ln(x) = 20. This is a significant step forward because we've successfully moved the constant term away from the logarithmic term. However, we're not quite there yet. The logarithmic term still has a coefficient (the number 4) that needs to be addressed. To fully isolate ln(x), we need to divide both sides of the equation by this coefficient. This process ensures that ln(x) stands alone, ready for the next stage of solving. By dividing both sides by 4, we effectively undo the multiplication and bring us closer to our solution. The resulting equation is ln(x) = 5, a much simpler form that clearly presents the relationship between the natural logarithm of x and the number 5. This isolation of the logarithmic term is the foundation upon which the rest of the solution will be built. It allows us to directly address the logarithm and, in the next step, eliminate it to reveal the value of x. This initial step is not just about manipulating the numbers; it's about strategically rearranging the equation to make it solvable. By isolating the logarithmic term, we've set the stage for the application of the inverse function, which will ultimately lead us to the solution. Remember, the key to solving complex equations often lies in breaking them down into simpler, more manageable steps, and isolating the key terms is a prime example of this principle.

2. Eliminating the Natural Logarithm

Now that we have successfully isolated the natural logarithm in our equation, which is now in the simplified form of ln(x) = 5, the next critical step is to eliminate this logarithm. This is where our understanding of inverse functions comes into play. The natural logarithm, denoted as 'ln', is essentially the logarithm to the base 'e', where 'e' is an irrational number approximately equal to 2.71828. The inverse operation of the natural logarithm is exponentiation with the base 'e'. In simpler terms, to undo the ln, we need to raise 'e' to the power of both sides of the equation. This is based on the fundamental mathematical principle that if ln(a) = b, then e^b = a. Applying this principle to our equation, we raise 'e' to the power of both ln(x) and 5. This gives us the equation e^(ln(x)) = e^5. On the left side of the equation, e^(ln(x)) simplifies beautifully to just x. This is because exponentiation and natural logarithm are inverse operations, meaning they effectively cancel each other out. So, we're left with x = e^5. This transformation is a crucial step because it frees 'x' from the confines of the logarithm, allowing us to directly calculate its value. The right side of the equation, e^5, represents a numerical value that we can approximate using a calculator or other computational tools. This step elegantly bridges the gap between the logarithmic form of the equation and the direct solution for 'x'. By understanding the inverse relationship between natural logarithms and exponentiation, we've been able to strategically manipulate the equation to isolate 'x'. This not only demonstrates a powerful problem-solving technique but also highlights the interconnectedness of mathematical concepts. Eliminating the logarithm is not just a mechanical process; it's a strategic maneuver that leverages the properties of inverse functions to simplify the equation and bring us closer to our final answer. In essence, this step embodies the core principle of undoing operations to isolate the variable we're trying to solve for. By carefully applying the inverse operation, we've transformed the equation into a form where 'x' is readily accessible.

3. Calculating the Approximate Value of x

Having successfully eliminated the natural logarithm and arrived at the equation x = e^5, the final step in solving for x is to calculate the approximate value of e^5. Here, we turn to our calculators or computational tools, as e is an irrational number, and its powers are best approximated. When you input e^5 into a calculator, you'll find that it yields a value approximately equal to 148.4131591. However, in the context of problem-solving, especially in multiple-choice scenarios or when dealing with real-world applications, we often need to round our answer to a reasonable level of precision. This is where understanding significant figures and the level of accuracy required becomes important. Looking at the answer choices provided in the original problem, we can see that they are rounded to varying degrees of precision. To match the format of the answer choices and provide a practical solution, we should round our calculated value of 148.4131591. If the choices are rounded to one decimal place, we would round our answer accordingly. In this case, rounding 148.4131591 to one decimal place gives us approximately 148.4. This rounding not only makes our answer consistent with the format of the given options but also reflects the fact that we're dealing with an approximation. The exact value of e^5 is an irrational number with an infinite number of decimal places, so in most practical situations, we use a rounded value. The process of calculating the approximate value of x is more than just punching numbers into a calculator; it's about understanding the nature of the numbers involved and presenting the answer in a meaningful and appropriate way. This step underscores the importance of both computational skills and the ability to interpret and present results effectively. By rounding our answer, we're not just simplifying the number; we're also communicating our solution in a way that is practical and relevant to the context of the problem. In the grand scheme of solving the equation, this final calculation brings closure to our efforts, providing a tangible answer that can be readily applied or interpreted.

4. Matching the Solution to the Options

After calculating the approximate value of x as 148.4, the final step in this problem-solving journey is to compare our calculated result with the provided answer options. This step is not merely about finding a matching number; it's about ensuring that our solution logically fits within the context of the problem and the given choices. It's a moment of verification where we confirm that our step-by-step process has led us to a sound conclusion. By carefully reviewing the answer options, we can identify the one that most closely aligns with our calculated value. In this case, the answer option D, which states x ≈ 148.4, perfectly matches our result. This alignment provides a strong sense of validation, assuring us that our approach was correct and our calculations were accurate. However, even if the match isn't immediately obvious, this step offers an opportunity to double-check our work. If our calculated value deviates significantly from all the options, it might indicate a potential error in our steps, prompting us to revisit our calculations or the logic of our solution. Matching the solution to the options also highlights the importance of understanding the level of precision required in a given problem. If the options are rounded to a specific decimal place, it reinforces the need for us to round our answer accordingly. This step emphasizes the practical aspect of problem-solving, where the final answer must not only be mathematically correct but also presented in a format that is meaningful and comparable to the available choices. In essence, this final comparison is a critical checkpoint in the problem-solving process. It's where we tie together our calculations, our understanding of the mathematical concepts, and the specific requirements of the question. By confidently matching our solution to the correct option, we demonstrate a comprehensive grasp of the problem and our ability to arrive at a logical and accurate answer. This step is not just the end of the solution; it's the culmination of a journey from the initial equation to a verified and validated answer.

Conclusion

In conclusion, solving the equation 4ln(x) - 8 = 12 involves a systematic approach of isolating the logarithmic term, eliminating the natural logarithm using its inverse operation, and calculating the approximate value of x. By following these steps carefully, we arrive at the solution x ≈ 148.4, which corresponds to option D. Remember, practice makes perfect, and the more you work with these types of equations, the more confident you'll become in solving them. For further learning and practice on logarithmic functions and equations, you can visit resources like Khan Academy.