Solving Exponential Equations A Step-by-Step Guide To 5e^(2x) - 4 = 11

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In this article, we will delve into solving exponential equations, focusing on the specific equation 5e^(2x) - 4 = 11. Exponential equations, which involve variables in the exponent, are fundamental in various fields such as mathematics, physics, engineering, and finance. Understanding how to solve them is crucial for tackling real-world problems. We will break down the steps involved in solving the given equation, providing a clear and concise explanation to enhance your understanding. This comprehensive guide aims to equip you with the knowledge and skills necessary to confidently solve similar exponential equations. By the end of this article, you will not only know the solution to this particular equation but also grasp the underlying principles applicable to a wider range of exponential problems.

Understanding Exponential Equations

Before diving into the solution, it’s important to understand what exponential equations are and why they are significant. Exponential equations are equations in which the variable appears in the exponent. These equations model many real-world phenomena, including population growth, radioactive decay, and compound interest. The general form of an exponential equation is ab^(cx) = d, where a, b, c, and d are constants, and x is the variable. The key to solving exponential equations lies in isolating the exponential term and then using logarithms to bring the variable down from the exponent. Logarithms are the inverse operation of exponentiation, and they allow us to solve for the variable x. Understanding the properties of exponents and logarithms is essential for solving these types of equations effectively. In this context, we will specifically address the equation 5e^(2x) - 4 = 11, demonstrating each step with detailed explanations to ensure clarity and comprehension. The importance of mastering exponential equations cannot be overstated, as they form the basis for more advanced mathematical concepts and applications in various scientific and financial disciplines.

Problem Statement: 5e^(2x) - 4 = 11

The problem we aim to solve is the exponential equation 5e^(2x) - 4 = 11. This equation involves the exponential function e raised to the power of 2x, where e is the base of the natural logarithm (approximately 2.71828). The equation is set up in such a way that we need to isolate the exponential term before we can apply logarithmic properties to solve for x. The presence of the constant terms and the coefficient of the exponential term necessitates a step-by-step approach to arrive at the correct solution. This particular equation is a classic example of the type of exponential equations encountered in algebra and calculus. Solving it involves a combination of algebraic manipulation and the application of logarithms, making it an excellent exercise for understanding these concepts. The equation's structure requires us to first isolate the exponential term, which involves adding and dividing to simplify the expression. Once the exponential term is isolated, we can use the natural logarithm to bring the exponent down and solve for x. The final solution will demonstrate how to use these principles effectively.

Step 1: Isolate the Exponential Term

The first step in solving the equation 5e^(2x) - 4 = 11 is to isolate the exponential term. This means we need to get the term containing e raised to a power by itself on one side of the equation. To do this, we start by adding 4 to both sides of the equation:

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

This simplifies to:

5e^(2x) = 15

Next, we divide both sides of the equation by 5 to completely isolate the exponential term:

5e^(2x) / 5 = 15 / 5

This further simplifies to:

e^(2x) = 3

Now that we have isolated the exponential term, we are ready to proceed to the next step, which involves using logarithms to solve for x. This isolation is a critical step because it allows us to apply the properties of logarithms correctly. Without isolating the exponential term, it would be difficult to apply logarithms effectively. This methodical approach ensures that we can accurately solve the equation by simplifying it step by step. The isolation of the exponential term is a common strategy in solving exponential equations, making it a crucial technique to master.

Step 2: Apply the Natural Logarithm

Once we have isolated the exponential term in the equation e^(2x) = 3, the next step is to apply the natural logarithm to both sides of the equation. The natural logarithm, denoted as ln, is the logarithm to the base e. Applying the natural logarithm helps us to bring the exponent down and solve for x. The natural logarithm is the inverse function of the exponential function with base e, which makes it the perfect tool for this situation. Applying the natural logarithm to both sides gives us:

ln(e^(2x)) = ln(3)

Using the property of logarithms that states ln(a^b) = b * ln(a), we can simplify the left side of the equation. Specifically, ln(e^(2x)) simplifies to 2x * ln(e). Since ln(e) is equal to 1, the equation becomes:

2x * 1 = ln(3)

Which simplifies to:

2x = ln(3)

This step is crucial because it transforms the exponential equation into a linear equation, which is much easier to solve. The application of the natural logarithm is a fundamental technique in solving exponential equations, and understanding how to use it is essential. This step allows us to manipulate the equation in a way that brings the variable out of the exponent, making it accessible for further algebraic manipulation.

Step 3: Solve for x

After applying the natural logarithm and simplifying the equation to 2x = ln(3), the final step is to solve for x. This involves isolating x on one side of the equation. To do this, we simply divide both sides of the equation by 2:

2x / 2 = ln(3) / 2

This gives us the solution:

x = ln(3) / 2

This is the exact solution for x. If we need an approximate numerical value, we can use a calculator to find the natural logarithm of 3 and then divide by 2. However, the exact solution is often preferred in mathematical contexts because it avoids rounding errors. The process of solving for x involves basic algebraic manipulation, which is a fundamental skill in mathematics. This final step underscores the importance of following the correct order of operations to arrive at the accurate solution. By isolating x, we have successfully solved the exponential equation. The ability to solve for variables in equations is a cornerstone of mathematical proficiency and is essential for various applications in science, engineering, and other fields.

Final Answer and Verification

The solution to the equation 5e^(2x) - 4 = 11 is x = ln(3) / 2. To verify this solution, we can substitute it back into the original equation and check if it holds true.

Substituting x = ln(3) / 2 into the original equation, we get:

5e^(2 * (ln(3) / 2)) - 4 = 11

Simplifying the exponent, we have:

5e^(ln(3)) - 4 = 11

Since e raised to the power of its natural logarithm cancels out, e^(ln(3)) simplifies to 3:

5 * 3 - 4 = 11

Performing the multiplication and subtraction:

15 - 4 = 11

11 = 11

Since the equation holds true, our solution x = ln(3) / 2 is verified. This verification step is crucial to ensure that the solution we obtained is correct. By substituting the solution back into the original equation, we confirm that it satisfies the equation's conditions. This process not only validates our answer but also reinforces our understanding of the steps involved in solving exponential equations. The final answer, therefore, is a definitive confirmation of our method and the accuracy of our calculations. In summary, the solution to the exponential equation 5e^(2x) - 4 = 11 is x = ln(3) / 2, which we have successfully verified.

Conclusion

In conclusion, we have successfully solved the exponential equation 5e^(2x) - 4 = 11 by following a systematic approach. We began by isolating the exponential term, then applied the natural logarithm to both sides of the equation to bring the exponent down. Finally, we solved for x and verified our solution. The key steps involved were:

  1. Isolating the exponential term by adding 4 to both sides and then dividing by 5, resulting in e^(2x) = 3.
  2. Applying the natural logarithm to both sides, which simplified to 2x = ln(3).
  3. Solving for x by dividing both sides by 2, giving us the final solution x = ln(3) / 2.
  4. Verifying the solution by substituting it back into the original equation to ensure it holds true.

This step-by-step process demonstrates a clear and effective method for solving exponential equations. Understanding and mastering these techniques is crucial for tackling more complex mathematical problems in various fields. Exponential equations are fundamental in many areas of science, engineering, and finance, and the ability to solve them accurately is a valuable skill. The solution x = ln(3) / 2 represents the exact value that satisfies the equation, and by verifying our solution, we have confirmed its correctness. This comprehensive approach not only provides the answer but also reinforces the underlying principles of exponential equations and their solutions.