Solving Exponential Equation $e^{x^2}=e^{12 X} \times \frac{1}{e^{32}}$ A Step-by-Step Guide

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In the realm of mathematics, solving exponential equations is a fundamental skill. This involves finding the values of the unknown variable that satisfy an equation where the variable appears in the exponent. Exponential equations pop up in various fields, including finance, physics, and computer science, making it crucial to master the techniques for solving them.

In this article, we will explore the methods for tackling a specific exponential equation. We'll start by introducing the core concepts behind exponential equations and then walk through the step-by-step solution process. By the end of this guide, you'll have a firm grasp of how to approach and solve similar problems.

Understanding Exponential Equations

Before we dive into solving the equation, it's essential to understand the fundamental principles of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. The general form of an exponential equation is:

af(x)=ag(x)a^{f(x)} = a^{g(x)}

Where:

  • 'a' is the base (a positive real number not equal to 1)
  • 'f(x)' and 'g(x)' are functions of x

To solve exponential equations, we aim to isolate the variable by using the properties of exponents and logarithms. One key property is that if the bases are equal, then the exponents must also be equal. This allows us to transform the exponential equation into a simpler algebraic equation.

Step-by-Step Solution

Let's tackle the specific equation given:

e^{x^2} = e^{12x} imes rac{1}{e^{32}}

Our goal is to find the values of 'x' that satisfy this equation. Here's a detailed walkthrough of the solution:

1. Simplify the Equation

The first step is to simplify the equation by using the properties of exponents. We can rewrite the equation as follows:

ex2=e12ximeseβˆ’32e^{x^2} = e^{12x} imes e^{-32}

When multiplying exponential terms with the same base, we add the exponents:

ex2=e12xβˆ’32e^{x^2} = e^{12x - 32}

2. Equate the Exponents

Now that we have the same base 'e' on both sides of the equation, we can equate the exponents:

x2=12xβˆ’32x^2 = 12x - 32

This transforms our exponential equation into a quadratic equation, which is much easier to solve.

3. Solve the Quadratic Equation

To solve the quadratic equation, we need to rearrange it into the standard form:

x2βˆ’12x+32=0x^2 - 12x + 32 = 0

We can solve this quadratic equation by factoring, using the quadratic formula, or completing the square. In this case, factoring is the simplest approach. We look for two numbers that multiply to 32 and add up to -12. These numbers are -4 and -8. Thus, we can factor the equation as:

(xβˆ’4)(xβˆ’8)=0(x - 4)(x - 8) = 0

Setting each factor equal to zero gives us the solutions:

xβˆ’4=0x - 4 = 0 or xβˆ’8=0x - 8 = 0

Solving for 'x' in each case, we get:

x=4x = 4 or x=8x = 8

4. Check the Solutions

It's always a good idea to check our solutions by plugging them back into the original equation. Let's check x = 4:

e^{(4)^2} = e^{12(4)} imes rac{1}{e^{32}}

e16=e48imeseβˆ’32e^{16} = e^{48} imes e^{-32}

e16=e48βˆ’32e^{16} = e^{48 - 32}

e16=e16e^{16} = e^{16}

This solution is valid. Now let's check x = 8:

e^{(8)^2} = e^{12(8)} imes rac{1}{e^{32}}

e64=e96imeseβˆ’32e^{64} = e^{96} imes e^{-32}

e64=e96βˆ’32e^{64} = e^{96 - 32}

e64=e64e^{64} = e^{64}

This solution is also valid.

Solution Set

Therefore, the solution set for the equation is:

{4, 8}

These are the values of 'x' that satisfy the original exponential equation.

Key Strategies for Solving Exponential Equations

To effectively solve exponential equations, keep these strategies in mind:

  • Simplify: Use the properties of exponents to simplify the equation.
  • Equate Exponents: If the bases are the same, equate the exponents.
  • Solve the Resulting Equation: You may end up with a linear, quadratic, or other type of equation. Use appropriate techniques to solve it.
  • Check Your Solutions: Always plug your solutions back into the original equation to ensure they are valid.

Advanced Techniques and Applications

While the basic approach works for many exponential equations, some problems require more advanced techniques. For instance, if the bases cannot be easily made the same, you might need to use logarithms. Taking the logarithm of both sides of the equation can help isolate the variable.

Exponential equations have wide-ranging applications in various fields:

  • Finance: Compound interest calculations involve exponential growth.
  • Physics: Radioactive decay and population growth are modeled using exponential functions.
  • Computer Science: Algorithms and data structures often have time complexities described by exponential functions.

Understanding how to solve exponential equations is crucial for tackling real-world problems in these areas.

Common Mistakes to Avoid

When working with exponential equations, it's important to avoid common pitfalls:

  • Incorrectly Applying Exponent Rules: Ensure you're using exponent rules correctly, especially when simplifying equations.
  • Forgetting to Check Solutions: Always verify your solutions to catch any extraneous roots.
  • Ignoring the Domain: Pay attention to the domain of exponential functions and logarithms, as some solutions may not be valid.

By being mindful of these mistakes, you can improve your accuracy and problem-solving skills.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. 32x=813^{2x} = 81
  2. 2x+1=322^{x+1} = 32
  3. 5x2βˆ’4x+3=15^{x^2 - 4x + 3} = 1

Work through these problems using the techniques discussed in this article. Check your answers to ensure you've mastered the concepts.

Conclusion

Solving exponential equations is a vital skill in mathematics and its applications. By understanding the core principles, using the right techniques, and avoiding common mistakes, you can confidently tackle a wide range of problems. In this article, we walked through a step-by-step solution to a specific exponential equation, highlighting the key strategies and considerations. Keep practicing, and you'll become proficient in solving exponential equations. This article equips you with the knowledge and skills needed to tackle exponential equations effectively. Remember to simplify equations, equate exponents, and solve resulting equations carefully. Always check your solutions to ensure accuracy. By mastering these techniques, you can confidently solve exponential equations in various contexts. Whether you're dealing with complex equations or basic problems, the principles remain the same. This knowledge is invaluable in fields like finance, physics, and computer science, where exponential functions play a crucial role. So, practice consistently and apply these strategies to excel in solving exponential equations. The ability to solve exponential equations is not just a mathematical skill; it's a powerful tool for understanding and solving real-world problems. From financial calculations to scientific models, exponential functions are everywhere. By mastering the techniques discussed in this article, you'll be well-equipped to handle these challenges. Remember to simplify expressions, equate the powers, and verify solutions to ensure accuracy. This comprehensive approach will help you tackle even the most complex exponential equations with confidence. Keep practicing and exploring the applications of these skills in various domains. With a solid understanding of exponential functions and solving techniques, you can unlock new possibilities in mathematics and beyond. This guide has provided a thorough foundation for solving exponential equations. Now it's up to you to apply this knowledge and continue your mathematical journey.