Solving E^(3x-2) = E^2 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. Exponential equations, a specific type of equation where the variable appears in the exponent, often present a unique challenge. This article delves into the process of solving the exponential equation e^(3x-2) = e^2, providing a detailed, step-by-step guide suitable for students and anyone interested in enhancing their mathematical abilities. Our primary keyword focus will be on solving exponential equations, ensuring this article serves as a valuable resource for those seeking clarity on the topic.
When you encounter an exponential equation like e^(3x-2) = e^2, the key is to understand the properties of exponents and logarithms. The goal is to isolate the variable, which in this case is x. Exponential equations are equations in which the variable appears in the exponent. To solve these types of equations, we often use properties of exponents and logarithms to manipulate the equation into a form where we can easily isolate the variable. The equation e^(3x-2) = e^2 is a classic example of an exponential equation that can be solved relatively easily by using the property that if a^m = a^n, then m = n, provided that the base a is the same on both sides of the equation. This property allows us to equate the exponents directly when the bases are equal, which significantly simplifies the process of finding the solution. Understanding these principles is crucial for tackling more complex exponential equations in the future.
To begin, we recognize that both sides of the equation have the same base, which is the natural exponential base e. This is a crucial observation because it allows us to apply a fundamental property of exponents: if a^m = a^n, then m = n. This property is the cornerstone of solving many exponential equations. In our case, a is e, m is (3x - 2), and n is 2. By applying this property, we can bypass the exponential nature of the equation and focus on a much simpler linear equation. This simplification is what makes solving this particular exponential equation straightforward. Recognizing this property is essential for anyone learning to solve exponential equations, as it transforms a seemingly complex problem into a manageable one. The ability to identify such opportunities is a key skill in mathematical problem-solving, allowing for efficient and accurate solutions. This initial step is often the most critical, setting the stage for the rest of the solution process.
Step 1: Equating the Exponents
Given the equation e^(3x-2) = e^2, we can directly equate the exponents due to the common base e. This step is justified by the one-to-one property of exponential functions. The one-to-one property of exponential functions states that if a^m = a^n, then m = n, where a is a positive number not equal to 1. This property is fundamental in solving exponential equations because it allows us to transform the equation from an exponential form to a simpler algebraic form. In our specific case, this means we can directly equate the exponents 3x - 2 and 2, resulting in the linear equation 3x - 2 = 2. This transformation is a critical step because it simplifies the problem significantly, allowing us to use basic algebraic techniques to solve for x. Without this property, solving exponential equations would be considerably more challenging. The one-to-one property provides a clear and direct path to isolating the variable, making it an indispensable tool in the arsenal of any mathematics student or professional.
This leads us to the equation:
3x - 2 = 2
This transformation is a significant simplification. We've moved from dealing with exponents to a simple linear equation, which is far easier to solve. The ability to make such transformations is a hallmark of effective mathematical problem-solving. By recognizing and applying the appropriate properties, we can break down complex problems into more manageable steps. The resulting equation, 3x - 2 = 2, is now within the realm of basic algebra, where we can use standard techniques to isolate x. This step highlights the importance of understanding the underlying principles of exponential functions and their properties. It's not just about memorizing formulas, but about understanding why they work and how to apply them in different situations. This deeper understanding allows for greater flexibility and confidence in solving mathematical problems.
Step 2: Solving the Linear Equation
Now we have a simple linear equation: 3x - 2 = 2. To solve for x, we need to isolate it on one side of the equation. The first step in isolating x is to add 2 to both sides of the equation. Adding the same value to both sides maintains the equality, which is a fundamental principle of algebraic manipulation. This operation effectively cancels out the -2 on the left side, moving us closer to isolating x. Understanding this principle is crucial for solving various types of equations, not just linear ones. The addition property of equality is a cornerstone of algebraic problem-solving, allowing us to manipulate equations while preserving their balance. This step is a clear demonstration of how basic algebraic principles can be applied to solve more complex problems, such as exponential equations. The process of isolating the variable is a core skill in mathematics, and this step provides a clear example of how it is applied.
Adding 2 to both sides, we get:
3x - 2 + 2 = 2 + 2 3x = 4
This simplifies the equation to 3x = 4. We are now one step closer to solving for x. The next step is to divide both sides of the equation by 3. Division, like addition, must be applied to both sides of the equation to maintain equality. This operation will isolate x on the left side, giving us the solution. The division property of equality is another fundamental principle in algebra, and its application here demonstrates its importance in solving equations. This step highlights the inverse relationship between multiplication and division, which is a key concept in mathematics. By understanding and applying these inverse operations, we can effectively solve for unknown variables in various equations. The ability to manipulate equations using these basic principles is essential for success in algebra and beyond.
To isolate x, we divide both sides by 3:
3x / 3 = 4 / 3 x = 4/3
Step 3: The Solution
Therefore, the solution to the equation e^(3x-2) = e^2 is x = 4/3. This is the value of x that satisfies the original equation. To verify the solution, we can substitute x = 4/3 back into the original equation and check if both sides are equal. This process of verification is a crucial step in problem-solving, as it ensures that the solution is correct and that no algebraic errors were made along the way. Substituting the solution back into the original equation provides a concrete check on the validity of the answer. It also reinforces the understanding of what it means to solve an equation – finding the value(s) of the variable(s) that make the equation true. This verification step is a valuable habit to develop in mathematics, as it promotes accuracy and confidence in one's work. It also helps to catch any potential mistakes and reinforces the problem-solving process.
Verification (Optional but Recommended)
To verify, substitute x = 4/3 into the original equation:
e^(3*(4/3) - 2) = e^2 e^(4 - 2) = e^2 e^2 = e^2
Since the left side equals the right side, our solution x = 4/3 is correct. This verification step confirms that our solution is indeed valid. The process of substitution is a powerful tool in mathematics, allowing us to check our work and ensure accuracy. In this case, it provides a clear and concise confirmation that x = 4/3 is the correct solution to the exponential equation e^(3x-2) = e^2. This step not only validates the solution but also reinforces the understanding of how the solution relates back to the original problem. It's a crucial part of the problem-solving process that should not be overlooked, as it adds a layer of certainty and reinforces the learning process.
In conclusion, solving the exponential equation e^(3x-2) = e^2 involves a few key steps: equating the exponents, solving the resulting linear equation, and verifying the solution. This example illustrates the power of understanding the properties of exponents and applying basic algebraic principles. Mastering these techniques is crucial for success in mathematics and related fields. Exponential equations appear in various contexts, from compound interest calculations to modeling population growth, making the ability to solve them a valuable skill. By following the steps outlined in this article, students and anyone interested in mathematics can confidently tackle similar problems. The key is to break down the problem into manageable steps, apply the appropriate properties and principles, and always verify the solution to ensure accuracy. This approach not only solves the immediate problem but also builds a strong foundation for future mathematical endeavors.
Remember, practice is key to mastering any mathematical skill. Work through various examples of exponential equations, and don't hesitate to review the fundamental concepts as needed. With consistent effort and a solid understanding of the underlying principles, you can confidently solve a wide range of exponential equations and excel in your mathematical studies. The journey to mathematical proficiency is a continuous one, and each solved problem adds to your expertise and confidence. Embrace the challenges, seek clarification when needed, and celebrate your successes along the way. The skills you develop in solving exponential equations will undoubtedly serve you well in various academic and professional pursuits.
