Solving For X In E^(3x+6) = 8 A Detailed Explanation
In this article, we will delve into the process of finding the value of x in the equation e^(3x+6) = 8. This equation involves an exponential function, and to solve for x, we will need to utilize the properties of logarithms, specifically the natural logarithm (ln). Understanding how to solve exponential equations is crucial in various fields, including mathematics, physics, engineering, and finance. This article will walk you through the steps to solve the equation, providing a detailed explanation of each step to ensure clarity and comprehension. This is a fundamental concept in algebra and calculus, so mastering this skill is beneficial for further studies in these areas. The ability to manipulate exponential and logarithmic functions is essential for solving many real-world problems, from population growth to radioactive decay.
We will begin by reviewing the basic properties of exponential functions and logarithms, focusing on the relationship between the exponential function e^x and the natural logarithm ln(x). The natural logarithm is the inverse function of the exponential function with base e, which means that ln(e^x) = x and e^(ln(x)) = x. This inverse relationship is key to solving exponential equations. By applying the natural logarithm to both sides of the equation, we can eliminate the exponential term and isolate the variable x. This technique is widely used in solving various mathematical problems and is a cornerstone of advanced mathematical concepts. The article aims to provide a comprehensive understanding of how to approach and solve such equations, making it a valuable resource for students and anyone interested in mathematics.
Understanding the properties of logarithms is crucial for solving exponential equations. Logarithms are the inverse operations of exponentiation. The logarithm of a number x with respect to a base b is the exponent to which b must be raised to produce x. In our case, we will use the natural logarithm, which has a base of e (Euler's number, approximately 2.71828). The natural logarithm is denoted as ln(x). The key property we'll use is that ln(e^y) = y. This property allows us to "undo" the exponential function. This inverse relationship is fundamental in solving exponential equations because it allows us to bring the exponent down and isolate the variable. Without this property, it would be much more difficult to solve equations where the variable is in the exponent. Therefore, a solid understanding of this property is essential for mastering exponential equations and logarithms.
Detailed Solution
Our goal is to find the value of x that satisfies the equation:
e^(3x+6) = 8
To isolate x, we'll use the natural logarithm.
Step 1: Apply the Natural Logarithm to Both Sides
Applying the natural logarithm (ln) to both sides of the equation, we get:
ln(e^(3x+6)) = ln(8)
Step 2: Simplify the Left Side
Using the property ln(e^y) = y, we can simplify the left side:
3x + 6 = ln(8)
Step 3: Isolate the Term with x
To isolate the term with x, subtract 6 from both sides:
3x = ln(8) - 6
Step 4: Solve for x
Finally, divide both sides by 3 to solve for x:
x = (ln(8) - 6) / 3
Answer
The value of x is:
x = (ln(8) - 6) / 3
Therefore, the correct answer is:
A. x = (ln(8) - 6) / 3
To reiterate the solution, let's break down each step with a more detailed explanation. We start with the equation e^(3x+6) = 8. Our primary objective is to isolate x. The first critical step involves applying the natural logarithm to both sides of the equation. This is done because the natural logarithm is the inverse function of the exponential function with base e. By applying the natural logarithm, we are essentially "undoing" the exponential operation, which allows us to bring the exponent down. This step is crucial because it transforms the equation from an exponential form to a linear form, which is much easier to solve. The application of the natural logarithm is based on the fundamental property that if a = b, then ln(a) = ln(b). This property is valid for any positive numbers a and b and is a cornerstone of solving exponential equations.
After applying the natural logarithm, we obtain ln(e^(3x+6)) = ln(8). The next step is to simplify the left side of the equation. Here, we use the crucial property of logarithms that ln(e^y) = y. This property states that the natural logarithm of e raised to any power is simply that power. Applying this property to our equation, we get 3x + 6 = ln(8). This simplification is the heart of the solution because it removes the exponential function, making the equation a simple linear equation in terms of x. Understanding this step is vital for mastering exponential equations, as it demonstrates how logarithms can be used to "undo" exponentiation.
Now that we have the equation 3x + 6 = ln(8), we need to isolate the term containing x. This is achieved by subtracting 6 from both sides of the equation. This operation maintains the equality and moves us closer to isolating x. The result is 3x = ln(8) - 6. This step is a standard algebraic manipulation, but it's essential for solving the equation. By isolating the term with x, we set the stage for the final step, which is to solve for x itself. The key here is to remember the basic principles of algebraic manipulation, ensuring that any operation performed on one side of the equation is also performed on the other side to maintain balance.
The final step in solving for x is to divide both sides of the equation by 3. This isolates x on one side, giving us the solution. Starting with 3x = ln(8) - 6, we divide both sides by 3 to get x = (ln(8) - 6) / 3. This is the final answer, expressing x in terms of the natural logarithm of 8. This step demonstrates the power of algebraic manipulation in solving equations. By carefully applying the rules of algebra, we can isolate the variable and find its value. The solution x = (ln(8) - 6) / 3 is the exact value of x that satisfies the original equation. It's important to note that this solution involves the natural logarithm, which is a transcendental number, meaning it cannot be expressed as a simple fraction or root. This highlights the importance of understanding and working with logarithmic functions in solving exponential equations.
When solving exponential equations like e^(3x+6) = 8, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy. One frequent error is misunderstanding the order of operations. In this specific case, it's crucial to apply the natural logarithm to both sides of the equation before attempting any other manipulations. Some students might try to divide or subtract before taking the logarithm, which is mathematically incorrect and will not lead to the correct solution. The logarithm must be applied first to "undo" the exponential function. This order is based on the inverse relationship between exponential and logarithmic functions, and deviating from this order will result in an incorrect solution.
Another common mistake is misapplying the properties of logarithms. For instance, students may incorrectly attempt to distribute the natural logarithm across terms within the parentheses, such as trying to say ln(3x + 6) = ln(3x) + ln(6), which is not true. The logarithm of a sum is not the sum of the logarithms. The correct property to use is ln(e^y) = y, which simplifies the equation after applying the natural logarithm to both sides. A clear understanding of logarithmic properties is essential to avoid such errors. These properties are fundamental rules that dictate how logarithms interact with different operations, and misapplying them can lead to significant errors in the solution.
Another pitfall is making algebraic errors during the simplification process. After applying the natural logarithm and simplifying to 3x + 6 = ln(8), students might make mistakes when isolating x. Common algebraic errors include incorrect subtraction or division, such as subtracting 6 from only one side of the equation or dividing only part of the expression by 3. It is crucial to perform the same operation on both sides of the equation to maintain equality and to ensure that all terms are correctly accounted for during the division step. These algebraic manipulations are the backbone of solving equations, and accuracy in these steps is critical for obtaining the correct answer.
Finally, a rounding error can occur if the student attempts to approximate ln(8) too early in the process. While calculators provide decimal approximations for logarithms, using these approximations prematurely can introduce errors that propagate through the rest of the solution. It's best to keep the exact value ln(8) until the final step, or at least carry several decimal places throughout the calculations to minimize rounding errors. This practice is particularly important in problems where precision is critical, and it helps ensure that the final answer is as accurate as possible. By avoiding these common mistakes, you can increase your chances of correctly solving exponential equations and similar problems.
In summary, solving the equation e^(3x+6) = 8 involves using the properties of logarithms, particularly the natural logarithm. The key steps include applying the natural logarithm to both sides, simplifying using the property ln(e^y) = y, and then isolating x through algebraic manipulation. The solution, x = (ln(8) - 6) / 3, demonstrates the importance of understanding the inverse relationship between exponential and logarithmic functions. This skill is fundamental in mathematics and has applications in various fields. By following the step-by-step approach outlined in this article, you can confidently solve similar exponential equations. The ability to manipulate these equations is crucial for more advanced mathematical concepts and real-world problem-solving.
Throughout this article, we have emphasized the importance of understanding each step in the solution process. From applying the natural logarithm to simplifying the equation and isolating the variable, each step plays a critical role in arriving at the correct answer. We have also highlighted common mistakes to avoid, such as misapplying logarithmic properties and making algebraic errors. By being mindful of these pitfalls, you can improve your accuracy and problem-solving skills. The concepts and techniques discussed here are not only applicable to this specific equation but also to a wide range of mathematical problems involving exponential and logarithmic functions. Mastering these skills will provide a solid foundation for further studies in mathematics and related fields.
In conclusion, solving exponential equations requires a systematic approach and a solid understanding of logarithmic properties. The equation e^(3x+6) = 8 serves as a valuable example of how to apply these principles to find the value of x. By following the detailed steps and avoiding common mistakes, you can confidently tackle similar problems and enhance your mathematical proficiency. The solution x = (ln(8) - 6) / 3 is not just an answer; it is a testament to the power of mathematical reasoning and the elegance of the relationship between exponential and logarithmic functions. This knowledge will serve you well in your future mathematical endeavors and beyond.
