Solving For X In 5^(x+2) = 5^9 A Step-by-Step Guide
Introduction
When dealing with exponential equations, understanding the fundamental principles of exponents is crucial. In this article, we will delve into solving the equation 5^(x+2) = 5^9, a classic example of an exponential equation where we need to find the value of x. Exponential equations are equations in which the variable appears in the exponent. They are used extensively in various fields such as physics, engineering, computer science, and finance, making the ability to solve them a valuable skill. We will explore the step-by-step process of solving this equation, highlighting the properties of exponents that allow us to simplify and find the solution. By the end of this discussion, you will have a clear understanding of how to tackle similar problems and a solid grasp of the underlying mathematical concepts.
Understanding Exponential Equations
Before we dive into the specific problem, let's briefly discuss exponential equations in general. An exponential equation is an equation in which the variable appears in one or more exponents. These equations often involve the same base raised to different powers. The key principle in solving such equations is that if the bases are the same, then the exponents must be equal for the equation to hold true. This principle allows us to simplify complex equations and solve for the unknown variable. Understanding this basic concept is essential for tackling a wide range of problems involving exponents. For instance, in real-world scenarios, exponential equations are used to model population growth, radioactive decay, and compound interest, among other phenomena. Therefore, mastering the techniques to solve these equations is not only academically beneficial but also practically useful.
Problem Statement: 5^(x+2) = 5^9
Now, let's focus on the problem at hand: Find the value of x if 5^(x+2) = 5^9. This equation presents a straightforward scenario where we can directly apply the principle of equating exponents. The equation has the same base (5) on both sides, which simplifies our task significantly. The problem is designed to test your understanding of how exponents work and your ability to apply the fundamental rules of algebra. Solving this equation will not only give us the value of x but also reinforce the method for solving other similar exponential equations. The beauty of this problem lies in its simplicity; it allows us to focus on the core concept without getting bogged down in complex calculations.
Step-by-Step Solution
To solve the equation 5^(x+2) = 5^9, we follow a simple yet effective approach.
Step 1: Equate the Exponents
Since the bases are the same (both sides have a base of 5), we can equate the exponents. This means that the expression in the exponent on the left side (x+2) must be equal to the exponent on the right side (9). This step is crucial because it transforms the exponential equation into a simple algebraic equation that is much easier to solve. The rule we're applying here is a fundamental property of exponential functions: if a^m = a^n, then m = n. By recognizing this property, we can bypass the complexities of dealing with exponents directly and focus on solving for the variable x.
Step 2: Form the Equation
Equating the exponents, we get the linear equation: x + 2 = 9. This equation is now a standard algebraic equation that can be solved using basic arithmetic operations. The transformation from the exponential equation to this linear equation is the key to simplifying the problem. It highlights the power of understanding the properties of exponents and how they can be used to manipulate equations. The simplicity of this equation allows us to isolate x quickly and accurately, leading us to the final solution.
Step 3: Solve for x
To solve for x, we need to isolate x on one side of the equation. We can do this by subtracting 2 from both sides of the equation: x + 2 - 2 = 9 - 2. This step maintains the equality of the equation while moving us closer to finding the value of x. The operation of subtracting 2 from both sides is a basic algebraic manipulation that ensures we are applying the same operation to both sides, preserving the balance. The resulting equation will directly give us the value of x.
Step 4: Final Answer
After performing the subtraction, we find that x = 7. This is the solution to the equation 5^(x+2) = 5^9. The value of x that satisfies the original equation is 7. We can verify this solution by substituting x = 7 back into the original equation: 5^(7+2) = 5^9, which simplifies to 5^9 = 5^9, confirming our solution. This final step of verification is crucial in mathematics to ensure the accuracy of our answer and to reinforce our understanding of the problem-solving process.
Answer: C. x = 7
Therefore, the correct answer is C. x = 7. This solution aligns with our step-by-step process, where we equated the exponents, formed the linear equation, and solved for x. The process demonstrates a clear and concise method for solving exponential equations with the same base. This problem serves as an excellent example of how understanding the fundamental properties of exponents can simplify complex equations. The ability to solve such equations is crucial in various fields of mathematics and science, making this a valuable skill to master.
Common Mistakes to Avoid
When solving exponential equations, there are several common mistakes students often make. Recognizing these potential pitfalls can help you avoid them and solve problems more accurately. One common mistake is incorrectly applying the properties of exponents. For example, some students might try to divide the bases instead of equating the exponents when the bases are the same. Another mistake is failing to simplify the equation before attempting to solve it, which can lead to unnecessary complexity. It’s crucial to always look for opportunities to simplify the equation first, such as combining like terms or using exponent rules to reduce the equation to its simplest form. Additionally, errors can occur when solving the resulting algebraic equation, such as making mistakes in arithmetic operations or not properly isolating the variable. To avoid these mistakes, it is essential to practice consistently, double-check your work, and ensure you have a solid understanding of the fundamental principles of algebra and exponents.
Practice Problems
To reinforce your understanding of solving exponential equations, here are a few practice problems:
- 3^(x-1) = 3^4
- 2^(2x+1) = 2^7
- 7^(x+3) = 7^5
Solving these problems will help solidify your skills and build confidence in tackling similar equations. Remember to follow the same step-by-step approach we used in the example: equate the exponents, form the equation, solve for x, and verify your solution. Practice is key to mastering any mathematical concept, and working through these problems will undoubtedly enhance your ability to solve exponential equations effectively. Moreover, these practice problems vary slightly in their structure, which will help you develop a flexible approach to problem-solving. By tackling these problems, you’ll be better prepared to handle more complex exponential equations in the future.
Conclusion
In conclusion, solving the equation 5^(x+2) = 5^9 demonstrates a fundamental principle in algebra: when the bases are the same, the exponents must be equal. By equating the exponents, we transformed the exponential equation into a simple algebraic equation, which we then solved to find that x = 7. This process highlights the importance of understanding the properties of exponents and how they can be applied to simplify equations. Avoiding common mistakes and practicing with similar problems will further enhance your problem-solving skills. Mastering these concepts not only helps in academic settings but also provides a solid foundation for tackling real-world problems involving exponential relationships. Remember, the key to success in mathematics is consistent practice and a thorough understanding of the underlying principles.
By following this detailed explanation and practicing regularly, you can confidently solve a wide range of exponential equations.
