Solving Exponential Equation 2e^(3x) = 400

In the realm of mathematics, exponential equations hold a significant place, often appearing in various scientific and engineering applications. These equations involve variables in the exponents, making their resolution a unique and crucial skill. This article delves into the process of solving an exponential equation, specifically 2e^(3x) = 400, where our goal is to isolate the variable x. We will explore the step-by-step method, utilizing the properties of logarithms and calculators to arrive at a precise solution, rounded to three decimal places. Understanding how to solve such equations is fundamental for anyone venturing into fields that rely on mathematical modeling and analysis.

Exponential equations are mathematical expressions where the variable appears in the exponent. These equations model various real-world phenomena, including population growth, radioactive decay, and compound interest. The general form of an exponential equation is a*b^(cx) = d, where a, b, c, and d are constants, and x is the variable we aim to solve. The key to solving these equations lies in understanding the inverse relationship between exponential and logarithmic functions. Logarithms allow us to "bring down" the exponent, making it possible to isolate the variable. For instance, the equation 2e^(3x) = 400 is a classic example of an exponential equation. Here, the variable x is in the exponent of the exponential function e^(3x). To find the value of x, we need to use logarithmic properties to undo the exponential operation. This typically involves taking the natural logarithm (ln) of both sides of the equation, since the base of the exponential function is e. However, before applying the logarithm, it's crucial to isolate the exponential term. In our case, this means dividing both sides of the equation by 2. Once the exponential term is isolated, we can apply the logarithm and use its properties to simplify the equation further. The understanding of exponential equations extends beyond mere algebraic manipulation. It involves grasping the underlying concepts of exponential growth and decay, which are essential in various scientific disciplines. Moreover, proficiency in solving these equations is not just a mathematical exercise; it's a tool for understanding and predicting real-world phenomena. For example, in finance, exponential equations are used to calculate compound interest, illustrating the power of exponential growth over time. Similarly, in biology, they are used to model population growth, providing insights into how populations change under different conditions. Therefore, mastering the techniques for solving exponential equations is an invaluable skill, equipping individuals with the ability to analyze and interpret a wide range of natural and mathematical phenomena.

To solve the equation 2e^(3x) = 400, we'll embark on a step-by-step journey, ensuring clarity and precision in each stage. This process not only provides the solution but also reinforces the fundamental principles of exponential equations. The first step in solving 2e^(3x) = 400 is to isolate the exponential term. This involves dividing both sides of the equation by the coefficient of the exponential term, which is 2 in this case. Dividing both sides by 2 gives us e^(3x) = 200. This step is crucial because it sets the stage for applying the logarithm, which is the key to unlocking the variable x from the exponent. Without isolating the exponential term, applying the logarithm becomes more complex and prone to errors. Next, we introduce the natural logarithm (ln) to both sides of the equation. The natural logarithm is the logarithm to the base e, which is the inverse function of the exponential function with base e. Applying the natural logarithm to both sides of e^(3x) = 200 yields ln(e^(3x)) = ln(200). The power of the natural logarithm lies in its ability to simplify exponential expressions. According to the properties of logarithms, ln(a^b) = b*ln(a). Applying this property to the left side of our equation, we get 3x*ln(e) = ln(200). Since ln(e) = 1, the equation further simplifies to 3x = ln(200). Now, the variable x is no longer in the exponent, making it easier to isolate. The next step is to isolate x by dividing both sides of the equation by 3. This gives us x = ln(200) / 3. At this point, we have an exact expression for x. However, to obtain a numerical value, we need to use a calculator. Using a calculator, we find that ln(200) ≈ 5.2983. Substituting this value into our equation, we get x ≈ 5.2983 / 3. Finally, we divide 5.2983 by 3 to get x ≈ 1.7661. The problem asks us to round our answer to three decimal places. Therefore, rounding 1.7661 to three decimal places gives us x ≈ 1.766. This is our final solution. In summary, solving the equation 2e^(3x) = 400 involves isolating the exponential term, applying the natural logarithm, using logarithmic properties to simplify the equation, and finally, using a calculator to obtain a numerical solution. Each step is crucial and builds upon the previous one, leading us to the final answer.

In solving exponential equations like 2e^(3x) = 400, the use of a calculator is often indispensable, especially when dealing with logarithms and exponential functions. A calculator not only speeds up the process but also ensures accuracy, particularly when dealing with irrational numbers like e and natural logarithms. To effectively use a calculator for this type of problem, familiarity with its functions is key. Most scientific calculators have dedicated buttons for natural logarithms (ln) and the exponential function (e^x). These functions are essential for solving exponential equations. After simplifying the equation to x = ln(200) / 3, the first step is to find the natural logarithm of 200. On a calculator, this typically involves pressing the "ln" button followed by entering "200" and pressing "enter" or "=". The calculator will display an approximate value for ln(200), which is approximately 5.2983. This value is an irrational number, meaning it has a non-repeating, non-terminating decimal representation. Therefore, it's crucial to use the calculator's full precision at this stage to minimize rounding errors in the final answer. Next, we divide the result by 3. This involves entering the division operation (÷) followed by "3" and pressing "enter" or "=". The calculator will then display the value of ln(200) / 3, which is approximately 1.7661. Now comes the crucial step of rounding the answer to the specified number of decimal places. In this case, we are asked to round to three decimal places. Rounding involves looking at the digit immediately to the right of the third decimal place. If this digit is 5 or greater, we round up the third decimal place. If it is less than 5, we leave the third decimal place as it is. In our case, the number is 1.7661. The digit to the right of the third decimal place (6) is 1, which is less than 5. Therefore, we round down, and the answer rounded to three decimal places is 1.766. It's important to note that rounding should be done only at the final step to avoid accumulating rounding errors. Rounding intermediate values can lead to a less accurate final answer. Calculators also have memory functions that can be used to store intermediate results, allowing for greater precision. In conclusion, using a calculator effectively involves understanding its functions, using it to obtain precise values for logarithms and exponential functions, and rounding the final answer to the specified number of decimal places. These skills are crucial for solving exponential equations accurately and efficiently.

In summary, solving the exponential equation 2e^(3x) = 400 requires a systematic approach that combines algebraic manipulation with the use of logarithms. By isolating the exponential term, applying the natural logarithm, and utilizing calculator functions, we can accurately determine the value of x. The key steps involve dividing both sides of the equation by 2, taking the natural logarithm of both sides, simplifying using logarithmic properties, and finally, using a calculator to evaluate the expression and round the answer to the desired precision. Through this process, we found that x ≈ 1.766. This solution highlights the practical application of exponential equations and the importance of understanding logarithms in mathematical problem-solving. Therefore, the final answer, rounded to three decimal places, is 1.766, which corresponds to option A).

Final Answer: A) 1.766