Solving Exponential Equations A Comprehensive Guide

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In the realm of mathematics, exponential equations present a fascinating challenge that combines the principles of exponents and algebra. These equations, where the variable appears in the exponent, require a unique set of techniques to unravel their solutions. This article delves into the intricacies of solving exponential equations, using the example equation 3(2^(4x)) = 21 as a case study. We will explore the fundamental concepts, step-by-step methods, and common pitfalls to avoid, equipping you with the skills to conquer these mathematical puzzles.

Understanding Exponential Equations

Exponential equations are equations where the variable appears in the exponent. They take the general form a^(f(x)) = b, where a is the base, f(x) is a function of the variable x, and b is a constant. Solving these equations involves isolating the exponential term and then using logarithms or other algebraic manipulations to find the value(s) of x that satisfy the equation.

Before diving into the solution process, it's crucial to grasp the core concepts underpinning exponential equations. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression 2^3, 2 is the base and 3 is the exponent, signifying 2 multiplied by itself three times (2 * 2 * 2 = 8). Exponential equations leverage this concept, but with the added complexity of the variable residing in the exponent.

Step-by-Step Solution: 3(2^(4x)) = 21

Let's tackle the equation 3(2^(4x)) = 21 step by step. This will provide a clear roadmap for solving similar exponential equations.

1. Isolate the Exponential Term

Our initial goal is to isolate the term containing the exponent. In this case, it's 2^(4x). To achieve this, we need to eliminate the coefficient 3 that's multiplying the exponential term. We can do this by dividing both sides of the equation by 3:

3(2^(4x)) / 3 = 21 / 3

This simplifies to:

2^(4x) = 7

Now, the exponential term is isolated on one side of the equation.

2. Apply Logarithms

Since we cannot easily express 7 as a power of 2, we introduce logarithms to solve for x. Logarithms are the inverse operation of exponentiation, allowing us to "bring down" the exponent.

We can apply the logarithm to both sides of the equation. The choice of logarithm base is flexible, but the common logarithm (base 10) or the natural logarithm (base e) are frequently used. For this example, let's use the natural logarithm (ln):

ln(2^(4x)) = ln(7)

A crucial property of logarithms comes into play here: ln(a^b) = b * ln(a). Applying this property, we can rewrite the left side of the equation:

4x * ln(2) = ln(7)

3. Solve for x

Now, we have a simple algebraic equation. To isolate x, we divide both sides by 4 * ln(2):

4x * ln(2) / (4 * ln(2)) = ln(7) / (4 * ln(2))

This simplifies to:

x = ln(7) / (4 * ln(2))

This is the exact solution for x. To obtain a numerical approximation, we can use a calculator:

x ≈ 0.7012

Therefore, the solution to the equation 3(2^(4x)) = 21 is approximately x = 0.7012.

Addressing the Multiple-Choice Question

Now, let's address the multiple-choice question associated with this equation:

Are the bases the same?

A. Yes, they can be written the same using division

B. Not enough information given

C. No

D. Yes

The bases in the equation 3(2^(4x)) = 21 are 2 (on the left side) and, implicitly, a base related to 21 on the right side. To determine if the bases are the same, we need to express both sides with the same base, if possible.

After isolating the exponential term, we have 2^(4x) = 7. Here, the base on the left side is 2, and the number on the right side is 7. It is not possible to directly express 7 as a power of 2 with a simple integer or fractional exponent. We needed to use logarithms to solve for x because 7 is not a power of 2. Therefore, the bases are fundamentally different in this context.

Therefore, the correct answer is C. No

The other options are incorrect because:

  • A. Yes, they can be written the same using division: This is misleading. While division is used in solving the equation, it doesn't make the bases the same. We cannot manipulate 2 and 7 through division to make them the same base in the context of exponents.
  • B. Not enough information given: We have a clear equation and have solved for x. There is sufficient information to determine that the bases are not directly the same.
  • D. Yes: This is incorrect because 7 cannot be expressed as a simple power of 2.

Key Concepts and Techniques

Let's solidify the key concepts and techniques involved in solving exponential equations:

  • Isolating the Exponential Term: The first step is always to isolate the exponential term on one side of the equation. This often involves basic algebraic operations like addition, subtraction, multiplication, or division.
  • Applying Logarithms: When the bases cannot be easily matched, logarithms are the go-to tool. Apply the logarithm to both sides of the equation. The choice of base (common or natural logarithm) is usually a matter of preference.
  • Logarithm Properties: Leverage the properties of logarithms, especially ln(a^b) = b * ln(a), to simplify the equation and bring the exponent down.
  • Solving for the Variable: After applying logarithms and simplifying, you'll typically have a linear equation in terms of the variable. Solve for the variable using standard algebraic techniques.

Common Pitfalls to Avoid

Solving exponential equations can be tricky, and certain pitfalls can lead to incorrect solutions. Here are some common mistakes to watch out for:

  • Incorrectly Applying Logarithms: Ensure you apply the logarithm to both sides of the equation and use the properties of logarithms correctly. Forgetting to apply the logarithm to a term or misusing the properties can lead to errors.
  • Dividing Before Isolating: Avoid dividing terms within the exponent before isolating the exponential term. This can complicate the equation unnecessarily.
  • Assuming Bases Can Always Be Matched: Not all numbers can be expressed as simple powers of the same base. Logarithms are essential when direct base matching isn't possible.
  • Forgetting the Order of Operations: Follow the correct order of operations (PEMDAS/BODMAS) when simplifying the equation.

Additional Examples and Practice

To further hone your skills, let's look at some additional examples of exponential equations:

  1. 5^(2x) = 125

    • Solution: 125 can be expressed as 5^3. Therefore, 5^(2x) = 5^3. Equating the exponents, we get 2x = 3, so x = 3/2.

  2. 2^(x+1) = 16

    • Solution: 16 can be expressed as 2^4. Therefore, 2^(x+1) = 2^4. Equating the exponents, we get x + 1 = 4, so x = 3.

  3. 4^(3x) = 8

    • Solution: Both 4 and 8 can be expressed as powers of 2 (4 = 2^2 and 8 = 2^3). Rewrite the equation as (22)(3x) = 2^3, which simplifies to 2^(6x) = 2^3. Equating the exponents, we get 6x = 3, so x = 1/2.

  4. 7^(x-2) = 49

    • Solution: 49 can be expressed as 7^2. Therefore, 7^(x-2) = 7^2. Equating the exponents, we get x - 2 = 2, so x = 4.

Conclusion

Solving exponential equations is a fundamental skill in mathematics with applications in various fields, including finance, physics, and computer science. By mastering the techniques outlined in this article, you can confidently tackle a wide range of exponential equations. Remember to isolate the exponential term, apply logarithms when necessary, and leverage the properties of logarithms to simplify the equation. With practice and attention to detail, you can conquer these mathematical challenges and unlock the power of exponential equations. Always double-check your solutions and be mindful of common pitfalls to ensure accuracy in your calculations. This comprehensive guide provides you with the tools and knowledge to excel in solving exponential equations.

Solve the equation 3(2^(4x)) = 21. Are the bases the same in this equation? Explain your reasoning.

Solving Exponential Equations Step-by-Step with Examples