Solving 2 + 3e^(x+2) = 7 For X A Step-by-Step Guide
In this article, we will delve into the process of solving the exponential equation 2 + 3e^(x+2) = 7 for the variable x. Exponential equations, where the variable appears in the exponent, are common in various fields such as mathematics, physics, engineering, and finance. Mastering the techniques to solve them is crucial for anyone working with these disciplines. We will break down the solution step by step, ensuring a clear understanding of each operation. Our goal is not only to find the value of x but also to illustrate the underlying principles of exponential equations and their solutions. This comprehensive guide will empower you to tackle similar problems with confidence. Understanding how to isolate the exponential term and then apply logarithms is key to solving these types of equations. Let's embark on this mathematical journey to uncover the value of x. This problem involves isolating the exponential term, which is a fundamental skill in solving many types of equations. By the end of this explanation, you should feel comfortable handling equations that involve exponential functions. The steps we take here will serve as a foundation for more complex problems in calculus and differential equations.
Problem Statement
We are given the exponential equation:
2 + 3e^(x+2) = 7
Our task is to find the value of x that satisfies this equation. This involves algebraic manipulation and the use of logarithms to isolate x. The equation combines basic arithmetic operations with an exponential function, making it a good example of how different mathematical concepts come together. Solving this equation requires a methodical approach, starting with simplifying the equation and then applying logarithmic properties. Understanding the properties of exponents and logarithms is essential for solving these types of problems. The solution will demonstrate how these properties can be used to isolate the variable and find its value.
Step-by-Step Solution
Step 1: Isolate the Exponential Term
Our first step is to isolate the exponential term, 3e^(x+2). To do this, we subtract 2 from both sides of the equation:
2 + 3e^(x+2) - 2 = 7 - 2
This simplifies to:
3e^(x+2) = 5
This step is crucial because we need the exponential term by itself before we can apply any further operations. Subtracting the constant term from both sides maintains the equation's balance while moving us closer to isolating the exponential term. This initial isolation sets the stage for the next steps, where we will divide and then apply logarithms. Isolating the term is a common strategy in solving various types of equations, not just exponential ones. The goal is always to get the term containing the variable by itself on one side of the equation. This is a foundational technique in algebra and is essential for solving more complex equations as well.
Step 2: Divide by the Coefficient
Next, we divide both sides of the equation by 3 to further isolate the exponential term e^(x+2):
(3e^(x+2)) / 3 = 5 / 3
This gives us:
e^(x+2) = 5/3
Dividing by the coefficient is another important step in isolating the exponential term. By performing this operation, we simplify the equation and make it ready for the application of logarithms. This step ensures that we are dealing directly with the exponential function and not a multiple of it. The fraction 5/3 is now the value that the exponential function must equal. This value will play a key role when we take the logarithm of both sides in the next step. This process of dividing by the coefficient is a fundamental algebraic technique that is widely used in solving equations of various forms. It helps to simplify the equation and isolate the variable or term of interest.
Step 3: Apply Natural Logarithm
To solve for x in 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 operation:
ln(e^(x+2)) = ln(5/3)
Using the property of logarithms that ln(e^a) = a, we simplify the left side:
x + 2 = ln(5/3)
The natural logarithm is the key to unlocking the exponent and solving for x. Applying the natural logarithm to both sides of the equation allows us to bring the exponent down as a coefficient. The property ln(e^a) = a is a fundamental identity in logarithmic functions and is essential for solving exponential equations. This step transforms the exponential equation into a linear equation, which is much easier to solve. Understanding and applying this property is crucial for anyone working with exponential and logarithmic functions. The choice of the natural logarithm is strategic here because it directly corresponds to the base of the exponential function in our equation, which is e. This simplification makes the next step straightforward.
Step 4: Solve for x
Finally, we solve for x by subtracting 2 from both sides of the equation:
x + 2 - 2 = ln(5/3) - 2
This gives us the solution:
x = ln(5/3) - 2
This step completes the process of isolating x and finding its value. By subtracting 2 from both sides, we obtain the final solution in terms of the natural logarithm. This solution is an exact value for x, and it can be approximated using a calculator if needed. The final solution highlights the importance of following each step carefully to arrive at the correct answer. Each step builds upon the previous one, leading us to the final result. This solution can be checked by substituting it back into the original equation to ensure that it satisfies the equation. The process of solving for x involves a series of algebraic manipulations and the application of logarithmic properties, demonstrating a systematic approach to problem-solving.
Final Answer
The solution to the equation 2 + 3e^(x+2) = 7 is:
x = ln(5/3) - 2
Therefore, the correct answer is:
C. x = ln(5/3) - 2
This final answer represents the value of x that makes the original equation true. The step-by-step solution we followed demonstrates a clear and logical approach to solving exponential equations. Understanding these steps will enable you to tackle similar problems with confidence. The final solution is expressed in terms of the natural logarithm, which is the most accurate way to represent the answer. If a numerical approximation is needed, a calculator can be used to evaluate ln(5/3). The entire process, from isolating the exponential term to applying logarithms and solving for x, showcases the interconnectedness of algebraic and logarithmic concepts. This final answer serves as a culmination of all the steps taken and confirms our solution process.
Conclusion
In this article, we have successfully solved the exponential equation 2 + 3e^(x+2) = 7 for x. We followed a systematic approach, which included isolating the exponential term, applying the natural logarithm, and solving for the variable. This detailed explanation provides a solid understanding of the process involved in solving exponential equations. The key steps included subtracting constants, dividing by coefficients, applying logarithms, and isolating the variable. Each step was carefully explained to ensure clarity and comprehension. The techniques used here are applicable to a wide range of exponential equations. Mastering these techniques is crucial for success in various mathematical and scientific fields. By understanding the properties of exponents and logarithms, we can confidently solve these types of problems. The final solution, x = ln(5/3) - 2, demonstrates the power of algebraic manipulation and logarithmic functions. This exercise highlights the importance of a step-by-step approach in solving mathematical problems. We hope this guide has been helpful in enhancing your understanding of exponential equations and their solutions. Remember to practice these techniques with other problems to further solidify your understanding. Solving exponential equations is a fundamental skill that is essential for various applications in mathematics, science, and engineering.
