Solving Exponential Equations A Detailed Guide To 4 + 5e^(x+2) = 11

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In the realm of mathematics, exponential equations play a crucial role in modeling various real-world phenomena, ranging from population growth and radioactive decay to compound interest and the spread of diseases. Solving these equations often requires a blend of algebraic manipulation and a solid understanding of exponential and logarithmic functions. This article delves into a step-by-step solution for the exponential equation 4 + 5e^(x+2) = 11, providing a clear and comprehensive guide for students, educators, and anyone interested in honing their mathematical skills. We will not only present the solution but also elucidate the underlying principles and techniques involved, ensuring a thorough understanding of the process. Let's embark on this mathematical journey and unravel the intricacies of solving exponential equations.

Understanding Exponential Equations

Before diving into the specifics of our equation, let's take a moment to define what exponential equations are and why they hold such significance in mathematics and its applications. An exponential equation is an equation in which the variable appears in the exponent. These equations are characterized by a constant base raised to a variable power, such as a^x = b, where 'a' and 'b' are constants and 'x' is the variable. The key to solving exponential equations lies in isolating the exponential term and then employing logarithms to bring the variable down from the exponent.

Exponential equations are not mere theoretical constructs; they are powerful tools for modeling a vast array of real-world phenomena. For instance, in biology, exponential functions describe the growth of populations under ideal conditions. In finance, they model the compounding of interest over time. In physics, they are used to describe the decay of radioactive substances. Understanding how to solve exponential equations is therefore essential for anyone seeking to apply mathematical principles to real-world problems.

Moreover, exponential equations serve as a foundation for understanding logarithmic functions, which are the inverses of exponential functions. Logarithms provide a way to solve for the exponent in an exponential equation, making them indispensable tools in various scientific and engineering disciplines. The ability to manipulate exponential and logarithmic expressions is a fundamental skill in advanced mathematics and its applications.

In the context of our specific equation, 4 + 5e^(x+2) = 11, we encounter a classic example of an exponential equation where the variable 'x' is nestled within the exponent. Our task is to isolate 'x' and determine its value. This will involve a series of algebraic steps, including isolating the exponential term, applying logarithms, and simplifying the resulting expression. By carefully navigating these steps, we will arrive at the solution and gain a deeper appreciation for the power and elegance of exponential equations.

Step-by-Step Solution of 4 + 5e^(x+2) = 11

Now, let's embark on the step-by-step solution of the equation 4 + 5e^(x+2) = 11. Our goal is to isolate 'x', and we'll achieve this by systematically unwrapping the equation, layer by layer.

Step 1: Isolate the Exponential Term

The first step in solving any exponential equation is to isolate the exponential term. In our case, the exponential term is 5e^(x+2). To isolate it, we need to get rid of the constant term, which is 4, on the left side of the equation. We can do this by subtracting 4 from both sides of the equation:

4 + 5e^(x+2) - 4 = 11 - 4

This simplifies to:

5e^(x+2) = 7

Now, we have the exponential term mostly isolated. There's still a coefficient of 5 multiplying it, so we need to eliminate that as well. We can do this by dividing both sides of the equation by 5:

(5e^(x+2)) / 5 = 7 / 5

This simplifies to:

e^(x+2) = 7/5

At this point, we have successfully isolated the exponential term. This is a crucial milestone because it allows us to proceed to the next step, which involves using logarithms to solve for 'x'.

Step 2: Apply Logarithms

The next step is to apply logarithms to both sides of the equation. The purpose of using logarithms is to bring the exponent down, making it easier to solve for 'x'. Since our exponential term has a base of 'e' (Euler's number), we'll use the natural logarithm (ln), which is the logarithm to the base 'e'. Applying the natural logarithm to both sides of the equation, we get:

ln(e^(x+2)) = ln(7/5)

The key property of logarithms that we'll use here is that ln(a^b) = b * ln(a). In our case, this means that ln(e^(x+2)) can be simplified to (x+2) * ln(e). Since ln(e) is equal to 1, we have:

(x+2) * 1 = ln(7/5)

This simplifies to:

x + 2 = ln(7/5)

Now, we have a simple linear equation in terms of 'x'. The logarithm has successfully brought the exponent down, and we are one step closer to finding the value of 'x'.

Step 3: Solve for x

To solve for 'x', we simply need to isolate it on one side of the equation. We can do this by subtracting 2 from both sides:

x + 2 - 2 = ln(7/5) - 2

This simplifies to:

x = ln(7/5) - 2

And there we have it! We have successfully solved for 'x'. The solution to the equation 4 + 5e^(x+2) = 11 is x = ln(7/5) - 2.

Verifying the Solution

In mathematics, it's always a good practice to verify your solution. This helps to ensure that you haven't made any errors in your calculations and that your solution is indeed correct. To verify our solution, we'll substitute x = ln(7/5) - 2 back into the original equation and see if it holds true.

The original equation is:

4 + 5e^(x+2) = 11

Substituting x = ln(7/5) - 2, we get:

4 + 5e^((ln(7/5) - 2) + 2) = 11

Simplifying the exponent, we have:

4 + 5e^(ln(7/5)) = 11

Using the property that e^(ln(a)) = a, we get:

4 + 5 * (7/5) = 11

Simplifying further:

4 + 7 = 11

11 = 11

As we can see, the equation holds true. This confirms that our solution, x = ln(7/5) - 2, is indeed correct. Verifying the solution is a crucial step in the problem-solving process, as it provides confidence in the accuracy of the result and helps to catch any potential errors.

Conclusion

In this article, we have embarked on a detailed exploration of solving the exponential equation 4 + 5e^(x+2) = 11. We began by understanding the fundamental nature of exponential equations and their significance in modeling real-world phenomena. We then meticulously walked through the step-by-step solution, which involved isolating the exponential term, applying logarithms, and solving for 'x'. Finally, we verified our solution to ensure its accuracy. The solution we arrived at is x = ln(7/5) - 2.

Solving exponential equations is a fundamental skill in mathematics, with applications spanning various fields, including science, engineering, and finance. 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 real-world problems. The key takeaway is the systematic approach to solving these equations: isolate the exponential term, apply logarithms to bring the exponent down, and then solve the resulting linear equation. Remember to always verify your solution to ensure its correctness.

Exponential equations may seem daunting at first, but with practice and a solid understanding of the underlying principles, they can be solved with confidence. This article has provided a comprehensive guide to solving one such equation, and we encourage you to apply these techniques to other exponential equations and further hone your mathematical skills. The world of mathematics is vast and fascinating, and the journey of exploration and discovery is a rewarding one.