Solving Exponential Equations A Step-by-Step Guide To 3125 = 5^(-10 + 3x)
Introduction to Exponential Equations
In the realm of mathematics, exponential equations hold a significant place, especially when dealing with growth, decay, and various other real-world phenomena. These equations involve variables in the exponents, making them distinct from polynomial equations. To master algebra and calculus, proficiency in solving exponential equations is crucial. Here, we aim to dissect and solve the exponential equation $3,125 = 5^{-10 + 3x}$, providing a comprehensive, step-by-step guide. This article will not only offer a solution to the given problem but also enhance your understanding of the underlying principles and methodologies for tackling similar equations. Before diving into the specifics, let's understand the core concepts that govern exponential equations.
At its core, an exponential equation is one where the variable appears in the exponent. Examples include , , and our featured equation, $3,125 = 5^{-10 + 3x}$. The key to solving these equations lies in manipulating them so that we can compare exponents directly. This typically involves expressing both sides of the equation using the same base. The base in an exponential expression is the number being raised to a power (e.g., in , the base is 5). The exponent is the power to which the base is raised (e.g., in , the exponent is 3).
The fundamental principle we use to solve exponential equations is that if , then , provided that is a positive number not equal to 1. This principle allows us to transform an exponential equation into a simpler algebraic equation by equating the exponents. The process involves several key steps:
- Express both sides of the equation with the same base.
- If the bases are the same, set the exponents equal to each other.
- Solve the resulting algebraic equation.
- Check your solution in the original equation.
In our case, we have $3,125 = 5^{-10 + 3x}$. The challenge is to rewrite 3,125 as a power of 5 so that we can apply this principle effectively. Let’s embark on this journey, breaking down each step with clarity and precision.
Step 1: Express Both Sides with the Same Base
In this initial step, our goal is to express both sides of the equation, $3,125 = 5^{-10 + 3x}$, using the same base. To achieve this, we need to recognize that 3,125 can be written as a power of 5. Often, this requires some trial and error or knowledge of common powers. Let’s break down the number 3,125 into its prime factors to see if it's indeed a power of 5.
We can start dividing 3,125 by 5:
- 3,125 ÷ 5 = 625
- 625 ÷ 5 = 125
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
From this division, we see that 3,125 can be expressed as 5 multiplied by itself five times, which means $3,125 = 5^5$. Now that we have this crucial piece of information, we can rewrite our original equation with both sides using the base of 5.
Our equation, $3,125 = 5^{-10 + 3x}$, now becomes $5^5 = 5^{-10 + 3x}$. This transformation is a critical step because it allows us to directly compare the exponents, thanks to the fundamental property of exponential equations that states if $a^m = a^n$, then $m = n$. By expressing both sides with the same base, we’ve set the stage for the next step, where we equate the exponents and solve for x. This method showcases the elegance of using exponent rules to simplify complex equations into more manageable forms. Understanding how to identify and convert numbers into their exponential forms with a common base is a foundational skill in solving exponential equations. Let's move on to the next step where we leverage this common base to find the value of x.
Step 2: Equate the Exponents
Having successfully expressed both sides of the equation $3,125 = 5^-10 + 3x}$ with the same base, we've arrived at the simplified form$. Now, we apply the core principle of solving exponential equations: if the bases are the same, then the exponents must be equal. This allows us to transition from an exponential equation to a simple algebraic equation.
By equating the exponents, we remove the exponential part and focus on the powers. The exponent on the left side of the equation is 5, and the exponent on the right side is -10 + 3x. Setting these equal to each other gives us the linear equation:
This equation is much easier to solve compared to the original exponential equation. We’ve effectively transformed the problem into a basic algebraic task. The significance of this step cannot be overstated; it is the bridge between the complexities of exponential equations and the straightforwardness of linear equations. By understanding and applying this principle, you can tackle a wide array of exponential problems with confidence.
This transformation highlights the power of algebraic manipulation. What initially seemed like a daunting task—solving an equation with a variable in the exponent—has been reduced to solving a simple linear equation. This process underscores a fundamental strategy in mathematics: break down complex problems into simpler, more manageable parts. Now that we have our linear equation, the next step involves isolating x to find its value. Let's proceed to the next section where we solve for x and complete our solution.
Step 3: Solve for x
With the exponential part of our equation behind us, we are now faced with a straightforward linear equation: $5 = -10 + 3x$. Solving for x involves isolating x on one side of the equation. This requires us to perform basic algebraic operations, ensuring we maintain the balance of the equation by doing the same operation on both sides.
Our first task is to eliminate the -10 from the right side of the equation. We can do this by adding 10 to both sides:
This simplifies to:
Now, to isolate x, we need to remove the coefficient 3. We can do this by dividing both sides of the equation by 3:
This simplifies to:
Thus, we find that $x = 5$. This is the solution to our algebraic equation, and, by extension, to our original exponential equation. The process of solving for x illustrates the importance of understanding and applying basic algebraic principles. Each step, from adding to dividing, is a testament to the power of systematic manipulation in mathematics. We have successfully isolated x and found its value, but our journey isn't quite over yet. The final step is to check our solution to ensure it satisfies the original equation. This verification step is crucial in mathematics, as it confirms the accuracy of our work and guards against potential errors. Let’s move on to the final step where we verify our solution.
Step 4: Check the Solution
The final step in solving any equation, especially exponential equations, is to check the solution. This crucial step ensures that the value we found for x actually satisfies the original equation. By substituting our solution back into the original equation, we can verify its correctness and catch any potential errors made during the solving process.
Our original equation was: $3,125 = 5^{-10 + 3x}$. We found that $x = 5$, so now we substitute this value back into the equation:
Now, let’s simplify the exponent on the right side:
So, our equation becomes:
We already know from our earlier steps that $5^5$ is indeed equal to 3,125. Therefore, our equation holds true:
This verification confirms that our solution, $x = 5$, is correct. The importance of this step cannot be overstated. Checking our solution provides a level of confidence in our answer and reinforces the problem-solving process. It's a testament to the rigorous nature of mathematics, where precision and accuracy are paramount.
In conclusion, we have successfully solved the exponential equation $3,125 = 5^{-10 + 3x}$ by following a systematic, step-by-step approach. We expressed both sides with the same base, equated the exponents, solved the resulting linear equation, and, most importantly, checked our solution. This comprehensive process not only gives us the correct answer but also deepens our understanding of the principles underlying exponential equations. This methodical approach can be applied to a wide range of mathematical problems, making it an invaluable skill for students and professionals alike. Solving exponential equations might seem challenging at first, but with practice and a clear understanding of the steps involved, you can confidently tackle these problems and expand your mathematical prowess.
Conclusion
In summary, solving the exponential equation $3,125 = 5^{-10 + 3x}$ involved a clear, step-by-step process:
- Express both sides of the equation with the same base.
- Equate the exponents.
- Solve for x.
- Check the solution.
By meticulously following these steps, we determined that $x = 5$ is the correct solution. This exercise not only provides the answer to the specific equation but also reinforces a general methodology applicable to solving other exponential equations. The key takeaway is the importance of breaking down complex problems into simpler, manageable steps. Each step, from expressing numbers as powers to solving linear equations, builds upon fundamental mathematical principles.
Furthermore, the act of checking the solution is a critical component of the problem-solving process. It ensures accuracy and reinforces understanding. Mathematics is not just about finding the answer; it’s about understanding the process and being confident in the result. This step-by-step guide serves as a valuable resource for anyone looking to enhance their mathematical skills, particularly in the realm of exponential equations.
Mastering these techniques opens doors to more advanced topics in mathematics, including calculus and differential equations, where exponential functions play a crucial role. As you continue your mathematical journey, remember that practice is key. The more you solve problems, the more comfortable and proficient you will become. Embrace the challenges, break down the complexities, and celebrate the solutions. Happy solving!
