Solving Exponential Equations A Step-by-Step Guide To 3,125 = 5^{-10 + 3x}

Introduction

In this article, we will delve into the process of solving an exponential equation. Exponential equations, characterized by variables appearing in the exponent, are a fundamental topic in mathematics with wide-ranging applications in fields like finance, physics, and computer science. The specific equation we will tackle is 3,125 = 5^{-10 + 3x}. Our goal is to find the value of x that satisfies this equation. To accomplish this, we will utilize the properties of exponents and logarithms, gradually isolating x and arriving at the solution. By understanding the steps involved in solving this equation, you'll gain valuable insights into handling exponential equations in general. This detailed exploration aims to provide a comprehensive understanding for anyone looking to enhance their mathematical skills.

Understanding Exponential Equations

Before diving into the solution, it's crucial to grasp the concept of exponential equations. An exponential equation is an equation where the variable appears in the exponent. For instance, in the equation 3,125 = 5^{-10 + 3x}, the expression -10 + 3x is the exponent. Solving such equations often involves manipulating the equation to isolate the variable. This typically requires leveraging the properties of exponents and logarithms. Understanding these properties is key to simplifying and solving exponential equations effectively. The ability to recognize and apply these properties is a fundamental skill in algebra and calculus, making this a crucial concept for any mathematics student or professional. To master this, it’s essential to practice with a variety of problems, gradually increasing in complexity. Remember, the core idea is to express both sides of the equation in terms of the same base, which simplifies the process of equating the exponents.

Breaking Down the Equation

Let's break down the equation 3,125 = 5^{-10 + 3x} step by step. First, we need to recognize that 3,125 can be expressed as a power of 5. Specifically, 3,125 = 5^5. This transformation is crucial because it allows us to have the same base on both sides of the equation. Once we have the same base, we can equate the exponents. This is a fundamental property of exponential equations: if a^m = a^n, then m = n. By applying this property, we can transform the exponential equation into a linear equation, which is much easier to solve. The process of finding the appropriate power of the base often involves prime factorization or recognizing common powers. This step is essential in simplifying the equation and setting the stage for the subsequent algebraic manipulations. Understanding and mastering this technique is critical for solving a wide range of exponential equations.

Step-by-Step Solution

Now, let’s walk through the step-by-step solution of the equation 3,125 = 5^{-10 + 3x}.

1. Rewrite 3,125 as a Power of 5: We know that 3,125 = 5^5. So, we can rewrite the equation as 5^5 = 5^{-10 + 3x}.
2. Equate the Exponents: Since the bases are the same, we can equate the exponents: 5 = -10 + 3x.
3. Isolate the Variable Term: Add 10 to both sides of the equation to isolate the term with x: 5 + 10 = -10 + 3x + 10, which simplifies to 15 = 3x.
4. Solve for x: Divide both sides by 3 to solve for x: 15 / 3 = 3x / 3, which gives us x = 5.

Thus, the solution to the equation 3,125 = 5^{-10 + 3x} is x = 5. This step-by-step approach demonstrates how to systematically solve an exponential equation by manipulating it into a simpler form and isolating the variable. Each step is crucial, and understanding the logic behind each manipulation is essential for solving more complex equations. This method provides a clear and concise way to tackle exponential equations, making it an invaluable tool in mathematical problem-solving.

Verification of the Solution

To ensure our solution is correct, we need to verify it. Verification involves substituting the value of x we found back into the original equation and checking if it holds true. In this case, we found that x = 5. So, we substitute x = 5 into the original equation 3,125 = 5^{-10 + 3x}.

Substituting x = 5, we get:

3,125 = 5^{-10 + 3(5)}

Simplify the exponent:

3,125 = 5^{-10 + 15}

3,125 = 5^5

Since 5^5 = 3,125, the equation holds true. This confirms that our solution x = 5 is correct. Verification is a critical step in solving any equation, as it helps to catch any potential errors made during the solving process. It provides confidence in the correctness of the solution and reinforces the understanding of the equation-solving process. By consistently verifying solutions, one can develop a habit of ensuring accuracy and avoiding mistakes in mathematical problem-solving. This step is particularly important in complex problems where errors can easily occur.

Alternative Methods

While we solved the equation by expressing both sides with the same base, there are alternative methods for solving exponential equations. One common method involves using logarithms. Logarithms are the inverse operation to exponentiation, and they can be particularly useful when it's difficult to express both sides of the equation with the same base. To solve the equation 3,125 = 5^{-10 + 3x} using logarithms, we could take the logarithm of both sides. For example, we could take the base-5 logarithm:

log₅(3,125) = log₅(5^{-10 + 3x})

Using the property of logarithms that logₐ(a^b) = b, we get:

log₅(3,125) = -10 + 3x

Since log₅(3,125) = 5, the equation simplifies to:

5 = -10 + 3x

This is the same equation we obtained earlier, and we can solve it as before to get x = 5. Using logarithms provides a versatile approach to solving exponential equations, especially when dealing with more complex problems. This method showcases the interconnectedness of different mathematical concepts and offers an alternative perspective on problem-solving. Understanding logarithms and their properties is a valuable skill for anyone studying mathematics or related fields.

Common Mistakes to Avoid

When solving exponential equations, several common mistakes can lead to incorrect solutions. Recognizing and avoiding these pitfalls is crucial for achieving accurate results. One frequent mistake is incorrectly applying the properties of exponents. For example, students might try to distribute an exponent over a sum or difference, which is not mathematically valid. Another common error is failing to express both sides of the equation with the same base before equating the exponents. This can lead to incorrect simplifications and ultimately, the wrong solution. Additionally, errors can occur when manipulating the equation algebraically, such as incorrectly adding or subtracting terms. To avoid these mistakes, it's essential to have a solid understanding of the properties of exponents and algebraic manipulations. Practice is key to mastering these concepts and developing the ability to recognize and correct errors. It's also helpful to double-check each step of the solution process to ensure accuracy. By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your ability to solve exponential equations correctly.

Conclusion

In conclusion, we have successfully solved the exponential equation 3,125 = 5^{-10 + 3x} and found that x = 5. We achieved this by expressing both sides of the equation with the same base, equating the exponents, and solving the resulting linear equation. We also verified our solution to ensure its correctness and discussed alternative methods using logarithms. Understanding how to solve exponential equations is a valuable skill in mathematics, with applications in various fields. By mastering the techniques discussed in this article and avoiding common mistakes, you can confidently tackle a wide range of exponential equation problems. Continuous practice and a solid understanding of the underlying principles are key to success in this area of mathematics. This comprehensive guide aims to equip you with the knowledge and skills necessary to excel in solving exponential equations.