Solving The Exponential Equation E^(-6x) = (e^8)^(x+14)

Introduction

In this article, we will delve into the process of solving exponential equations, focusing on the specific equation $e^{-6x} = (e^8)^{x+14}$. Exponential equations, characterized by variables appearing in the exponents, are a fundamental concept in mathematics with applications across various fields, including physics, engineering, and finance. Understanding how to solve these equations is crucial for anyone pursuing studies or careers in these areas. This comprehensive guide will walk you through the necessary steps, providing a clear and concise explanation of the underlying principles. We will break down the equation, apply relevant exponential rules, and systematically isolate the variable to find the solution. By the end of this article, you will have a solid understanding of how to tackle similar exponential equations with confidence. Let's embark on this mathematical journey together, demystifying the complexities of exponential equations and empowering you with the skills to solve them effectively. The techniques we explore here are not only applicable to this particular problem but also provide a foundation for addressing a wide range of exponential equations. So, whether you are a student preparing for an exam or a professional seeking to refresh your knowledge, this article serves as a valuable resource. We will emphasize clarity and precision in our explanations, ensuring that each step is easily understood. Furthermore, we will highlight common pitfalls and provide tips for avoiding errors, thereby enhancing your problem-solving abilities. Get ready to sharpen your mathematical skills and conquer the world of exponential equations!

Understanding Exponential Equations

Before we dive into solving the equation, it's essential to understand the basic principles of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. These equations are governed by the properties of exponents, which dictate how powers behave under various operations. For example, the power of a power rule states that $(a^m)^n = a^{mn}$, which will be crucial in simplifying our equation. Another fundamental property is that if $a^m = a^n$, then $m = n$, provided that $a$ is a positive number not equal to 1. This property allows us to equate exponents once we have the same base on both sides of the equation. Understanding these rules is the bedrock for solving any exponential equation. We must also be familiar with the exponential function, denoted as $e^x$, where 'e' is Euler's number, an irrational constant approximately equal to 2.71828. This function plays a significant role in many mathematical and scientific applications, and equations involving 'e' often arise in natural phenomena. In our case, the equation involves powers of 'e', making it a classic example of an exponential equation. To solve it effectively, we need to apply the properties of exponents and logarithms strategically. Recognizing the structure of an exponential equation and the rules that govern it is the first step towards finding a solution. Without a solid grasp of these fundamentals, tackling more complex equations becomes significantly more challenging. Therefore, we will take the time to thoroughly review these concepts before proceeding with the solution. This foundational knowledge will not only help us solve the given equation but also equip us to handle a wider variety of problems involving exponents and exponential functions.

Step 1: Simplify the Equation

The given equation is $e^{-6x} = (e^8)^{x+14}$. The first step in simplifying the equation is to apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule to the right side of the equation, we get:

$e^{-6x} = e^{8(x+14)}$

This step is crucial because it allows us to express both sides of the equation with the same base, which is 'e' in this case. Once we have the same base on both sides, we can equate the exponents, as we will see in the next step. Simplifying the equation in this manner makes it much easier to work with and reduces the complexity of the problem. The power of a power rule is a fundamental concept in algebra, and its application here demonstrates its importance in solving exponential equations. By correctly applying this rule, we transform the equation into a more manageable form, setting the stage for further steps towards the solution. This initial simplification is often the key to unlocking the solution, as it allows us to focus on the exponents rather than the exponential terms themselves. Without this step, the equation would be significantly more challenging to solve. Therefore, it is essential to master this rule and its application to effectively tackle exponential equations. Remember, the goal is to manipulate the equation in a way that isolates the variable, and this simplification is a major step in that direction. By transforming the right side of the equation, we have brought it into a form that is directly comparable to the left side, paving the way for the next phase of the solution.

Step 2: Equate the Exponents

Now that we have $e^{-6x} = e^{8(x+14)}$, we can equate the exponents. This is a valid step because if $a^m = a^n$, then $m = n$, provided that $a$ is a positive number not equal to 1. In our case, the base is 'e', which satisfies this condition. Therefore, we can set the exponents equal to each other:

$-6x = 8(x + 14)$

This equation is now a linear equation in terms of 'x', which is much easier to solve compared to the original exponential equation. Equating the exponents is a powerful technique that allows us to transform a complex equation into a simpler one. This step is the heart of solving exponential equations where the bases are the same. By focusing on the exponents, we eliminate the exponential terms and deal with a straightforward algebraic equation. It is crucial to understand the logic behind this step to effectively solve such equations. Without equating the exponents, we would be stuck with a more challenging problem. This technique is a direct consequence of the properties of exponents and logarithms, and it is a cornerstone of exponential equation solving. By recognizing that the bases are the same, we can confidently equate the exponents and proceed with solving the resulting linear equation. This step highlights the importance of understanding the fundamental principles of algebra and their application to more complex problems. The ability to equate exponents is a valuable skill that will serve you well in various mathematical contexts. By mastering this technique, you will be able to tackle a wide range of exponential equations with greater ease and confidence. This step is not just a mechanical procedure; it is a logical deduction based on the properties of exponential functions, and understanding this logic is key to becoming a proficient problem solver.

Step 3: Solve for x

We now have the linear equation $-6x = 8(x + 14)$. To solve for x, we first distribute the 8 on the right side:

$-6x = 8x + 112$

Next, we want to isolate the 'x' terms on one side of the equation. We can do this by adding $6x$ to both sides:

$0 = 14x + 112$

Now, subtract 112 from both sides:

$-112 = 14x$

Finally, divide both sides by 14 to find the value of x:

$x = -8$

Thus, we have found the solution to the equation. Solving for 'x' involves a series of algebraic manipulations designed to isolate the variable. Each step is a logical operation that preserves the equality of the equation. Distributing the 8 is a crucial step in simplifying the equation and removing the parentheses. Adding $6x$ to both sides allows us to consolidate the 'x' terms on one side, making it easier to isolate 'x'. Subtracting 112 from both sides further isolates the 'x' term, bringing us closer to the solution. Finally, dividing by 14 gives us the value of 'x'. This process demonstrates the power of algebraic techniques in solving equations. By applying these techniques systematically, we can unravel complex equations and find their solutions. It is essential to be meticulous and careful with each step to avoid errors. Double-checking your work is always a good practice to ensure accuracy. The ability to solve for 'x' is a fundamental skill in algebra, and it is essential for tackling a wide range of mathematical problems. By mastering this skill, you will be well-equipped to solve more complex equations and tackle challenging mathematical concepts. This step-by-step approach highlights the importance of breaking down a problem into smaller, manageable steps, making the solution process more accessible and less daunting. With practice, you will become more proficient at solving for 'x' and tackling a variety of algebraic equations with confidence.

Step 4: Verification

To ensure the solution is correct, we need to verify it by substituting $x = -8$ back into the original equation:

$e^{-6(-8)} = (e^8)^{-8+14}$

$e^{48} = (e^8)^6$

$e^{48} = e^{8 imes 6}$

$e^{48} = e^{48}$

The left side of the equation equals the right side, confirming that our solution, $x = -8$, is correct. Verification is a crucial step in the problem-solving process. It allows us to catch any errors that might have occurred during the solution process and ensures that we have arrived at the correct answer. By substituting the solution back into the original equation, we are essentially performing a check to see if the equation holds true. If the left side of the equation equals the right side, then the solution is verified. If not, it indicates that there might be an error in our calculations, and we need to go back and review our steps. In this case, the verification process confirms that $x = -8$ is indeed the correct solution. This step provides peace of mind and strengthens our confidence in the answer. It also reinforces the importance of accuracy and attention to detail in mathematical problem-solving. Verification is not just a formality; it is an essential part of the process that ensures the validity of our solution. By making it a habit to verify our solutions, we can avoid making mistakes and develop a deeper understanding of the problem-solving process. This step also helps us to build our problem-solving skills and become more confident in our ability to tackle mathematical challenges. It is a testament to the rigor and precision that are inherent in mathematics, where every step must be justified and verified to ensure accuracy.

Conclusion

In conclusion, we have successfully solved the exponential equation $e^{-6x} = (e^8)^{x+14}$ and found the solution to be $x = -8$. We accomplished this by applying the properties of exponents, equating the exponents, solving the resulting linear equation, and verifying the solution. Understanding exponential equations and the techniques for solving them is essential in various fields of mathematics and science. This step-by-step guide provides a clear and concise approach to tackling such problems. By following these steps, you can confidently solve similar exponential equations. This process highlights the importance of understanding the underlying principles of mathematics and their application to problem-solving. Each step in the solution process is based on a fundamental mathematical concept, and by mastering these concepts, we can tackle more complex problems with greater ease and confidence. The ability to solve exponential equations is a valuable skill that will serve you well in various mathematical contexts. It is also a testament to the power of mathematics to provide solutions to real-world problems. By practicing and applying these techniques, you can enhance your problem-solving abilities and deepen your understanding of mathematics. This journey through the solution of an exponential equation demonstrates the beauty and elegance of mathematical reasoning, where each step logically follows from the previous one, leading to a clear and concise solution. It also underscores the importance of perseverance and attention to detail in mathematical problem-solving. With practice and dedication, you can master the art of solving exponential equations and unlock the door to a deeper understanding of mathematics.