Solving 7-3e^{-x}=2 Step-by-Step With Rounding To Nearest Hundredth
Introduction
In this comprehensive article, we will delve into the step-by-step process of solving the exponential equation 7-3e^{-x}=2. Our main objective is to isolate the variable x and determine its value, rounding the final answer to the nearest hundredth. Exponential equations like this one frequently appear in various fields, including mathematics, physics, engineering, and finance. Understanding how to solve them is a crucial skill for students and professionals alike. We will explore the properties of exponential functions, logarithms, and algebraic manipulations necessary to arrive at the solution. Let's embark on this mathematical journey together and demystify the process of solving exponential equations.
Understanding Exponential Equations
To effectively solve the exponential equation, it's crucial to understand the fundamental concepts underlying these equations. Exponential equations involve a variable in the exponent, making them distinct from linear or polynomial equations. The key to solving them lies in our ability to isolate the exponential term and then use logarithms to bring the exponent down. In our specific equation, 7-3e^{-x}=2, the term e^{-x} is the exponential term we need to isolate. The constant e represents the base of the natural logarithm, approximately equal to 2.71828. This constant plays a significant role in calculus and various scientific applications. Understanding the behavior of exponential functions, including their growth and decay patterns, is essential for solving these types of equations. Moreover, recognizing the relationship between exponential functions and their inverse logarithmic functions is paramount. We will utilize these relationships extensively in the subsequent steps to find the value of x.
Step-by-Step Solution
Let's break down the solution to the equation 7-3e^{-x}=2 into manageable steps:
1. Isolate the Exponential Term
Our initial goal is to isolate the term containing the exponent, which is e^{-x}. To achieve this, we start by subtracting 7 from both sides of the equation:
7 - 3e^{-x} - 7 = 2 - 7
-3e^{-x} = -5
Next, we divide both sides by -3:
e^{-x} = 5/3
Now, the exponential term e^{-x} is isolated on one side of the equation. This is a critical step as it sets the stage for using logarithms.
2. Apply Natural Logarithm
To remove the exponent, we apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the inverse function of the exponential function with base e, which allows us to "undo" the exponential. Applying the natural logarithm, we get:
ln(e^{-x}) = ln(5/3)
Using the property of logarithms that ln(a^b) = b*ln(a), we simplify the left side:
-x * ln(e) = ln(5/3)
Since ln(e) = 1, the equation becomes:
-x = ln(5/3)
3. Solve for x
To find x, we multiply both sides of the equation by -1:
x = -ln(5/3)
This gives us the exact solution for x in terms of the natural logarithm. However, to get a numerical approximation, we proceed to the next step.
4. Calculate the Numerical Value
Using a calculator, we compute the value of -ln(5/3):
x ≈ -0.5108
5. Round to the Nearest Hundredth
The problem requests that we round the answer to the nearest hundredth. Thus, we look at the thousandth place (the third digit after the decimal point) to determine whether to round up or down. In this case, the thousandth digit is 0, so we round down:
x ≈ -0.51
Therefore, the solution to the equation 7-3e^{-x}=2, rounded to the nearest hundredth, is approximately -0.51.
Importance of Logarithms
Logarithms play a pivotal role in solving exponential equations. They act as the inverse function to exponentiation, effectively "undoing" the exponential operation. In our example, the natural logarithm (ln) was crucial because it's the inverse of the exponential function with base e. Applying the natural logarithm to both sides of the equation allowed us to bring the exponent down, transforming the equation into a more manageable form. Without logarithms, it would be extremely challenging, if not impossible, to isolate the variable x in an exponential equation. The properties of logarithms, such as ln(a^b) = b*ln(a), are indispensable tools in this process. Understanding and utilizing logarithms is a fundamental skill for anyone dealing with exponential functions and equations in mathematics, science, and engineering.
Common Mistakes to Avoid
When solving exponential equations, several common mistakes can lead to incorrect answers. Recognizing and avoiding these pitfalls is essential for accuracy. One frequent error is incorrectly applying the order of operations. For example, in the equation 7-3e^{-x}=2, one might mistakenly subtract 3 from 7 before addressing the exponential term. Remember, multiplication and exponentiation take precedence over addition and subtraction. Another common mistake involves mishandling the negative sign. In our solution, we had to carefully account for the negative sign in the exponent and when isolating x. A sign error can completely change the result. Additionally, it's crucial to correctly apply the properties of logarithms. Forgetting or misapplying logarithmic identities can lead to incorrect simplification and ultimately an incorrect solution. Always double-check each step and pay close attention to details to minimize the risk of errors. Practicing a variety of problems and reviewing your work can help reinforce your understanding and prevent these common mistakes.
Real-World Applications
Exponential equations are not just abstract mathematical constructs; they have numerous real-world applications across various disciplines. In physics, exponential functions describe radioactive decay, where the amount of a radioactive substance decreases exponentially over time. In biology, they model population growth, where a population can increase exponentially under ideal conditions. In finance, exponential functions are used to calculate compound interest, where the value of an investment grows exponentially over time. In engineering, they appear in the analysis of electrical circuits and the design of control systems. For instance, the charging and discharging of a capacitor in an RC circuit can be modeled using an exponential equation. Understanding how to solve these equations is therefore crucial for anyone working in these fields. The ability to model and predict real-world phenomena using exponential functions makes this a valuable skill.
Conclusion
In this article, we have thoroughly explored the process of solving the exponential equation 7-3e^{-x}=2 and rounding the solution to the nearest hundredth. We began by understanding the fundamental concepts of exponential equations and the importance of isolating the exponential term. We then meticulously walked through each step of the solution, from applying the natural logarithm to calculating the numerical value and rounding the final answer. We also emphasized the critical role of logarithms in solving exponential equations and discussed common mistakes to avoid. Furthermore, we highlighted the numerous real-world applications of exponential equations across various fields. By mastering the techniques presented in this article, you will be well-equipped to tackle a wide range of exponential equations and apply them to practical problems in mathematics, science, and engineering. The ability to confidently solve these equations is a valuable asset in both academic and professional pursuits.
