Solving Exponential Equations A Step-by-Step Guide

In this comprehensive guide, we will delve into the methods for solving exponential equations. Exponential equations are equations where the variable appears in the exponent. These types of equations are fundamental in various fields, including mathematics, physics, engineering, and finance. Mastering the techniques to solve these equations is crucial for understanding and applying exponential functions in real-world scenarios. We will explore two distinct exponential equations, each requiring a unique approach to find the solution. Our primary focus will be on the step-by-step methodologies to tackle these equations effectively. This article aims to equip you with the necessary skills and knowledge to confidently solve exponential equations. By understanding the properties of exponents and applying appropriate algebraic techniques, you can navigate through complex problems and arrive at accurate solutions. Whether you are a student learning about exponential functions for the first time or a professional seeking to refresh your knowledge, this guide will provide valuable insights and practical strategies. We will begin by examining the first equation, which involves manipulating exponents and using properties of logarithms. Then, we will move on to the second equation, which requires a substitution method to transform it into a more manageable form. Throughout the guide, we will emphasize clarity and precision in each step, ensuring that you can follow the logic and replicate the process. By the end of this guide, you will have a solid understanding of how to solve a variety of exponential equations and be well-prepared to tackle more advanced problems in this area.

Solving exponential equations often involves manipulating the bases to be the same. Let's tackle the first equation: 49^{x2} = 1/7^{3x-2}. The key to solving this equation lies in recognizing that both 49 and 1/7 can be expressed as powers of 7. Specifically, 49 is 7 squared (7^2), and 1/7 is 7 to the power of -1 (7^{-1}). By rewriting the equation in terms of a common base, we can simplify it and ultimately solve for x. The process begins by substituting 7^2 for 49 and 7^{-1} for 1/7. This transformation allows us to express the entire equation in terms of a single base, which is essential for equating the exponents. Once we have a common base, we can use the properties of exponents to further simplify the equation. For instance, when raising a power to another power, we multiply the exponents. This property is crucial in simplifying the left side of the equation, where we have (7²⁾{x^2}. Similarly, the right side of the equation can be simplified by applying the power rule. After simplifying both sides, we will have an equation where the bases are the same, and the exponents can be directly compared. This step is where the core of solving exponential equations becomes apparent: if the bases are equal, then the exponents must also be equal for the equation to hold true. By equating the exponents, we transform the exponential equation into a more manageable algebraic equation, typically a quadratic equation in this case. Solving the quadratic equation will give us the possible values of x that satisfy the original exponential equation. These solutions can then be verified by substituting them back into the original equation to ensure they are valid. This methodical approach of reducing the equation to a common base, simplifying using exponent rules, and solving the resulting algebraic equation is a fundamental technique in solving exponential equations. It is a technique that can be applied to a wide range of problems, making it an essential tool in your mathematical toolkit. By mastering this method, you will be well-equipped to tackle more complex exponential equations and related problems.

2.1 Step-by-Step Solution

1. Rewrite the equation using the base 7:
   • 49^{x2} can be rewritten as (7²⁾{x^2}
   • 1/7^{3x-2} can be rewritten as 7^{-(3x-2)}
   • The equation becomes: (7²⁾{x^2} = 7^{-(3x-2)}
2. Simplify the exponents:
   • Using the power rule (a^b)c = a^{b*c}, simplify the left side:
     • (7²⁾{x^2} = 7^{2x2}
   • The equation now is: 7^{2x2} = 7^{-(3x-2)}
3. Equate the exponents:
   • Since the bases are equal, the exponents must be equal:
     • 2x^2 = -(3x - 2)
4. Rearrange the equation into a quadratic equation:
   • Expand and move all terms to one side:
     • 2x^2 = -3x + 2
     • 2x^2 + 3x - 2 = 0
5. Solve the quadratic equation:
   • Factor the quadratic:
     • (2x - 1)(x + 2) = 0
   • Set each factor equal to zero and solve for x:
     • 2x - 1 = 0 => x = 1/2
     • x + 2 = 0 => x = -2
6. Solutions:
   • The solutions to the equation are x = 1/2 and x = -2.

Solving exponential equations like 9^x - 2(3^x) = 12 often involves recognizing a hidden quadratic form. This particular equation might appear complex at first glance, but by employing a clever substitution, we can transform it into a more familiar and manageable quadratic equation. The key observation here is that 9^x can be expressed as (3²⁾x, which is equivalent to (3^x)2. This insight allows us to see the equation as a quadratic expression in terms of 3^x. To make this transformation explicit, we introduce a substitution: let y = 3^x. By substituting y into the original equation, we replace 3^x with y and (3^x)2 with y^2. This results in a quadratic equation in the variable y, which we can then solve using standard methods such as factoring, completing the square, or using the quadratic formula. Once we have solved for y, we must remember that our ultimate goal is to find the values of x. Therefore, we need to reverse the substitution and replace y with 3^x. This gives us one or more exponential equations of the form 3^x = some value. To solve these equations, we can use logarithms. Specifically, we can take the logarithm of both sides of the equation, which allows us to isolate x. The choice of logarithm base is arbitrary, but using the natural logarithm (base e) or the common logarithm (base 10) is often convenient. After applying the logarithm, we can solve for x by dividing both sides of the equation by the logarithm of the base (in this case, log(3)). It is important to note that not all solutions for y will yield valid solutions for x. Since 3^x is always positive, any negative values of y will not result in real solutions for x. Therefore, we must check the solutions we obtain for y and discard any that are negative. This step-by-step process of recognizing the quadratic form, making a suitable substitution, solving the resulting quadratic equation, and then reversing the substitution to find the solutions for the original variable is a powerful technique in solving a wide range of exponential equations. By mastering this method, you will be able to approach these types of problems with confidence and efficiency.

3.1 Step-by-Step Solution

1. Recognize the quadratic form:
   • Rewrite 9^x as (3^x)2:
     • (3^x)2 - 2(3^x) = 12
2. Substitute y = 3^x:
   • Replace 3^x with y:
     • y^2 - 2y = 12
3. Rearrange into a quadratic equation:
   • Move all terms to one side:
     • y^2 - 2y - 12 = 0
4. Solve the quadratic equation:
   • Factor the quadratic:
     • (y - 6)(y + 2) = 0
   • Set each factor equal to zero and solve for y:
     • y - 6 = 0 => y = 6
     • y + 2 = 0 => y = -2
5. Substitute back 3^x for y:
   • 3^x = 6 and 3^x = -2
6. Solve for x:
   • For 3^x = 6:
     • Take the logarithm of both sides (using base 3 or natural logarithm):
       • x = log_3(6) or x = ln(6)/ln(3)
   • For 3^x = -2:
     • Since 3^x is always positive, there is no real solution for this case.
7. Solution:
   • The solution to the equation is x = log_3(6) or x ≈ 1.6309.

In conclusion, solving exponential equations requires a combination of algebraic manipulation, recognition of underlying structures, and a solid understanding of exponential and logarithmic properties. We have explored two distinct types of exponential equations in this guide, each demanding a unique approach. The first equation, 49^{x2} = 1/7^{3x-2}, was solved by expressing both sides with a common base, equating exponents, and solving the resulting quadratic equation. This method underscores the importance of recognizing equivalent expressions and utilizing the properties of exponents to simplify complex equations. The solutions obtained, x = 1/2 and x = -2, demonstrate the practical application of these techniques. The second equation, 9^x - 2(3^x) = 12, presented a different challenge. By recognizing the hidden quadratic form and employing a substitution, we transformed the equation into a more manageable quadratic equation in terms of a new variable, y. Solving for y and then reversing the substitution allowed us to find the solution for x. However, it was crucial to check the validity of the solutions, as the nature of exponential functions dictates that 3^x cannot be negative. This highlights the importance of considering the domain and range of functions when solving equations. The solution, x = log_3(6), showcases the application of logarithms in solving exponential equations, especially when a simple base manipulation is not sufficient. Throughout this guide, we have emphasized the importance of a step-by-step approach, breaking down complex problems into smaller, more manageable steps. This methodical approach is essential for accuracy and clarity in problem-solving. By mastering the techniques discussed in this guide, you will be well-equipped to tackle a wide range of exponential equations and related problems. Furthermore, the skills and concepts learned here extend beyond mathematics, finding applications in various scientific and engineering disciplines. Whether you are a student, educator, or professional, a solid understanding of exponential equations is a valuable asset in today's world.