Solving 5e^(2x) - 4 = 11 A Step-by-Step Guide
#h1 Solving Exponential Equations A Detailed Guide to 5e^(2x) - 4 = 11
In this article, we will delve into the step-by-step solution of the exponential equation 5e^(2x) - 4 = 11. Exponential equations are a crucial part of mathematics, often appearing in various fields such as physics, engineering, and finance. Understanding how to solve them is essential for anyone studying these disciplines. This guide provides a comprehensive breakdown of the solution process, ensuring clarity and accuracy at each step. By the end of this article, you will not only know the correct answer but also understand the underlying principles that make it so. So, let's dive into the world of exponential equations and master the techniques needed to solve them effectively.
Understanding Exponential Equations
Before we jump into solving the specific equation 5e^(2x) - 4 = 11, it's crucial to understand the nature of exponential equations. Exponential equations are those in which the variable appears in the exponent. The general form of an exponential equation is a^(f(x)) = b, where a and b are constants and f(x) is a function of x. Solving such equations involves isolating the exponential term and then using logarithms to bring the variable down from the exponent.
Exponential functions are used to model a variety of phenomena, including population growth, radioactive decay, and compound interest. The exponential function e^x is particularly important in calculus and its applications, where e is the base of the natural logarithm, approximately equal to 2.71828. The inverse of the exponential function e^x is the natural logarithm, denoted as ln(x). This relationship is key to solving exponential equations involving e.
The properties of logarithms are fundamental in solving exponential equations. The key property we will use in solving our equation is that ln(a^b) = b ln(a). This property allows us to bring the exponent b down as a coefficient, which is essential when the variable x is in the exponent. Additionally, understanding the relationship between exponential and logarithmic functions—specifically that e^(ln(x)) = x and ln(e^x) = x—is crucial for simplifying and solving exponential equations.
In the context of real-world applications, exponential equations can help us model and understand various scenarios. For example, in finance, the formula for compound interest involves an exponential term, allowing us to calculate the future value of an investment. In physics, radioactive decay follows an exponential decay model, where the amount of a radioactive substance decreases exponentially over time. These applications highlight the importance of mastering exponential equations for problem-solving in numerous fields.
Step-by-Step Solution of 5e^(2x) - 4 = 11
To solve the equation 5e^(2x) - 4 = 11, we will follow a series of algebraic steps to isolate x. Each step is crucial in ensuring we arrive at the correct solution. Let's break down the process:
Step 1: Isolate the Exponential Term
The first step in solving the equation is to isolate the exponential term. This means we want to get the term 5e^(2x) by itself on one side of the equation. To do this, we add 4 to both sides of the equation:
5e^(2x) - 4 + 4 = 11 + 4
This simplifies to:
5e^(2x) = 15
Now, we need to further isolate the exponential term by dividing both sides by 5:
(5e^(2x)) / 5 = 15 / 5
This gives us:
e^(2x) = 3
Step 2: Apply the Natural Logarithm
Once the exponential term is isolated, the next step is to apply the natural logarithm (ln) to both sides of the equation. This is because the natural logarithm is the inverse function of the exponential function with base e, which allows us to remove the exponential term:
ln(e^(2x)) = ln(3)
Using the property of logarithms that ln(e^a) = a, we can simplify the left side of the equation:
2x = ln(3)
Step 3: Solve for x
The final step is to solve for x. We do this by dividing both sides of the equation by 2:
(2x) / 2 = ln(3) / 2
This gives us the solution:
x = ln(3) / 2
Therefore, the solution to the equation 5e^(2x) - 4 = 11 is x = ln(3) / 2. This step-by-step approach ensures that we correctly apply the properties of exponential and logarithmic functions to find the solution.
Identifying the Correct Option
After solving the equation 5e^(2x) - 4 = 11, we found that x = ln(3) / 2. Now, let's compare this solution to the given options to identify the correct one. The options provided are:
- A. x = ln(3)
- B. x = ln(27)
- C. x = ln(3) / 2
- D. x = 3 / ln(3)
By directly comparing our solution x = ln(3) / 2 with the given options, it's clear that option C, x = ln(3) / 2, matches our solution. Therefore, option C is the correct answer.
It's essential to carefully match the derived solution with the provided options to avoid any errors. Sometimes, solutions can be expressed in different forms, and it may be necessary to simplify or rewrite the derived solution to match one of the options. However, in this case, the solution we found directly corresponds to option C, making it straightforward to identify the correct answer.
Common Mistakes and How to Avoid Them
When solving exponential equations like 5e^(2x) - 4 = 11, several common mistakes can lead to incorrect answers. Recognizing these mistakes and understanding how to avoid them is crucial for mastering exponential equations. Let's discuss some of the most frequent errors and the strategies to prevent them.
Mistake 1: Incorrectly Isolating the Exponential Term
One of the most common errors is failing to correctly isolate the exponential term. This often involves incorrectly applying the order of operations or making algebraic mistakes when moving terms across the equation. For instance, some might try to divide by 5 before adding 4, which would lead to an incorrect setup.
How to Avoid: Always follow the correct order of operations (PEMDAS/BODMAS) when isolating the exponential term. In our equation, this means first adding 4 to both sides and then dividing by 5. Double-check each step to ensure accuracy.
Mistake 2: Forgetting to Apply the Natural Logarithm to All Terms
Another frequent mistake is applying the natural logarithm only to one side of the equation or to some terms but not others. Remember, the logarithm must be applied to both sides of the equation to maintain equality.
How to Avoid: Ensure that the natural logarithm is applied to the entire expression on both sides of the equation. In the case of e^(2x) = 3, this means applying ln to both sides, resulting in ln(e^(2x)) = ln(3).
Mistake 3: Misapplying Logarithmic Properties
Incorrectly using logarithmic properties is another common pitfall. For example, not understanding that ln(e^a) = a can lead to errors when simplifying the equation. Similarly, mistakes in using the properties of logarithms to combine or separate terms can lead to wrong solutions.
How to Avoid: Review and understand the fundamental properties of logarithms. Specifically, remember that ln(e^a) = a, and be careful when using other properties such as ln(ab) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Apply these properties correctly and consistently.
Mistake 4: Algebraic Errors in Simplification
Algebraic mistakes, such as incorrect division or simplification, can occur at any stage of the problem. These errors can lead to a wrong final answer, even if the initial steps were correct.
How to Avoid: Be meticulous with each algebraic step. Double-check your work and simplify expressions carefully. If possible, use a calculator or software to verify calculations, especially when dealing with complex numbers or expressions.
Mistake 5: Not Checking the Final Answer
Finally, one of the most critical mistakes is not verifying the final solution. Plugging the solution back into the original equation can help identify errors made during the solving process.
How to Avoid: Always substitute the solution back into the original equation to check its validity. If the solution does not satisfy the original equation, there is likely an error in the steps, and you should review your work.
By being aware of these common mistakes and following the strategies to avoid them, you can improve your accuracy and confidence in solving exponential equations.
Conclusion
In conclusion, solving the exponential equation 5e^(2x) - 4 = 11 involves a series of precise steps that, when followed correctly, lead to the accurate solution. We began by isolating the exponential term, then applied the natural logarithm to both sides of the equation to simplify it. This allowed us to solve for x, ultimately finding that x = ln(3) / 2. By carefully comparing our solution with the given options, we confidently identified option C as the correct answer.
Throughout the solution process, we highlighted the importance of understanding the fundamental properties of exponential and logarithmic functions. The relationship between the exponential function e^x and the natural logarithm ln(x) is crucial in solving exponential equations. Moreover, we discussed common mistakes that students often make, such as incorrectly isolating terms or misapplying logarithmic properties, and provided strategies to avoid these errors.
Mastering exponential equations is essential for various fields, including mathematics, physics, engineering, and finance. The ability to solve these equations allows for the modeling and understanding of real-world phenomena, such as population growth, radioactive decay, and financial investments. By practicing and understanding the techniques outlined in this guide, you can improve your problem-solving skills and gain confidence in tackling more complex mathematical challenges.
Remember, the key to success in solving exponential equations lies in a clear understanding of the underlying principles and a meticulous approach to each step. By avoiding common mistakes and consistently practicing, you can develop the skills necessary to excel in this area of mathematics.
