Finding The Value Of K In Exponential Equations A Step By Step Solution

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Introduction: Deciphering Exponential Equations

In the realm of mathematics, exponential equations form a cornerstone of various calculations and problem-solving scenarios. These equations, characterized by the presence of variables in the exponents, often appear complex at first glance. However, with a solid grasp of exponential properties and algebraic manipulation techniques, these seemingly intricate problems can be systematically solved. In this article, we delve into a specific exponential equation, aiming to unveil the value of an unknown exponent, denoted as k. Our journey involves dissecting the equation, applying fundamental exponential rules, and ultimately arriving at the solution. We will explore the equation: 10^-3 ⋅ 10 ⋅ 10^k = 10^-3, where our primary objective is to determine the numerical value of k that satisfies this equation. This exploration will not only provide the answer to this particular problem but also reinforce the understanding of exponential properties, which are essential tools in various mathematical and scientific domains.

At the heart of solving exponential equations lies the understanding of exponential properties, which dictate how exponents behave under different mathematical operations. One of the most fundamental properties is the product of powers property, which states that when multiplying exponential expressions with the same base, we can add the exponents. Mathematically, this property is expressed as: a^m ⋅ aⁿ = a^m+n, where 'a' is the base and 'm' and 'n' are the exponents. Another crucial property is the negative exponent property, which defines how to deal with negative exponents. It states that a base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. This can be written as: a^-n = 1/aⁿ. These properties, along with others like the power of a power property and the quotient of powers property, form the arsenal for simplifying and solving exponential equations. In the context of our problem, we will primarily utilize the product of powers property to combine the exponential terms on the left side of the equation, thereby simplifying the equation and making it easier to isolate the unknown exponent k.

Before diving into the step-by-step solution, let's take a moment to appreciate the significance of exponents in mathematics and various scientific fields. Exponents provide a concise way to represent repeated multiplication, making it easier to handle very large or very small numbers. In computer science, exponents play a pivotal role in representing binary data, where powers of 2 are used extensively. In physics, exponents are used to describe phenomena that vary exponentially, such as radioactive decay or population growth. In finance, compound interest calculations rely heavily on exponential functions. Furthermore, exponents are fundamental to scientific notation, a standard way of expressing numbers in a compact form. The power of exponents extends beyond mere mathematical notation; they are a powerful tool for modeling and understanding the world around us. Thus, mastering exponential properties and solving exponential equations is not just an academic exercise but a valuable skill applicable across various disciplines. As we embark on solving for k in the given equation, we are not merely finding a numerical answer but also reinforcing our understanding of these essential mathematical concepts.

Problem Statement: Isolating the Unknown Exponent

The core of our task lies in determining the value of k in the equation: 10^-3 ⋅ 10 ⋅ 10^k = 10^-3. This equation, an example of an exponential equation, presents a unique challenge due to the unknown exponent k. The key to solving this problem is to isolate k by leveraging the fundamental properties of exponents. These properties provide us with the tools to simplify the equation, combine terms, and ultimately reveal the value of k that satisfies the equation. The equation itself consists of exponential terms with a common base, which is 10. This common base is crucial because it allows us to apply the product of powers property, which we discussed earlier. This property will enable us to combine the exponential terms on the left side of the equation into a single exponential term, making the equation more manageable.

Breaking down the components of the equation, we observe that we have three exponential terms on the left side: 10^-3, 10, and 10^k. The term 10^-3 represents 10 raised to the power of -3, which, according to the negative exponent property, is equivalent to 1/10³ or 0.001. The term 10 is simply 10 raised to the power of 1 (though the exponent 1 is often implicitly understood and not explicitly written). The term 10^k is where our unknown k resides. On the right side of the equation, we have 10^-3, which is the same as the first term on the left side. Our goal is to manipulate the left side of the equation using exponential properties until we have a single term with a base of 10 and an exponent that includes k. This will then allow us to equate the exponents on both sides of the equation and solve for k.

The strategy for solving this equation involves a series of logical steps. First, we will apply the product of powers property to combine the exponential terms on the left side. This means adding the exponents of the terms being multiplied. Once we have simplified the left side to a single exponential term, we will have an equation of the form 10^exponent = 10^-3. At this point, we can invoke the property that if two exponential expressions with the same base are equal, then their exponents must also be equal. This allows us to set the exponent on the left side equal to -3 and solve the resulting algebraic equation for k. This step-by-step approach, guided by the properties of exponents, will lead us to the solution. By carefully applying these properties and techniques, we can confidently unveil the value of k that satisfies the given equation. The problem-solving process not only yields the answer but also deepens our understanding of how exponents behave and how they can be manipulated to solve mathematical problems.

Step-by-Step Solution: Applying Exponential Properties

To embark on our step-by-step solution, let's revisit the original equation: 10^-3 ⋅ 10 ⋅ 10^k = 10^-3. The first critical step in solving this equation is to simplify the left side by applying the product of powers property. As we discussed earlier, this property states that when multiplying exponential expressions with the same base, we add the exponents. In our case, the common base is 10, and the exponents we need to add are -3, 1 (since 10 is equivalent to 10¹), and k. Therefore, applying the product of powers property, we can rewrite the left side of the equation as 10^{-3 + 1 + k}.

Now, let's simplify the exponent by combining the numerical terms. We have -3 + 1, which equals -2. So, the exponent becomes -2 + k. Our equation now looks like this: 10^{-2 + k} = 10^-3. This simplified form is a significant milestone in our solution process because it brings us closer to isolating k. The equation now presents a clear comparison of two exponential expressions with the same base. This is crucial because it allows us to leverage another fundamental property of exponents: If two exponential expressions with the same base are equal, then their exponents must also be equal. This property forms the bridge between the exponential world and the algebraic world, allowing us to translate the exponential equation into a simpler algebraic equation.

With the equation in the form 10^{-2 + k} = 10^-3, we can now equate the exponents. This means setting the exponent on the left side, which is -2 + k, equal to the exponent on the right side, which is -3. This gives us the algebraic equation: -2 + k = -3. This is a simple linear equation that we can solve for k using basic algebraic techniques. To isolate k, we need to get rid of the -2 on the left side. We can do this by adding 2 to both sides of the equation. This maintains the balance of the equation and allows us to isolate k. Adding 2 to both sides, we get: -2 + k + 2 = -3 + 2. This simplifies to k = -1. And there we have it, the solution to our problem. The value of k that satisfies the original exponential equation is -1. This result can be verified by substituting -1 for k in the original equation and confirming that both sides are indeed equal.

Solution Verification: Ensuring Accuracy

Having arrived at a solution, it's crucial to verify its accuracy. This step is not just a formality but a vital part of the problem-solving process. Verification ensures that we haven't made any errors in our calculations or application of exponential properties. In the context of our problem, verifying the solution involves substituting the value we found for k back into the original equation and checking if the equation holds true. This process provides a concrete confirmation that our solution is correct and that we have a thorough understanding of the problem.

Let's revisit the original equation: 10^-3 ⋅ 10 ⋅ 10^k = 10^-3. Our solution is that k = -1. To verify this, we substitute -1 for k in the equation: 10^-3 ⋅ 10 ⋅ 10^-1 = 10^-3. Now, we need to simplify the left side of the equation and see if it equals the right side. We can apply the product of powers property again, adding the exponents on the left side. We have -3, 1 (from the 10), and -1. Adding these together, we get -3 + 1 + (-1) = -3. So, the left side of the equation simplifies to 10^-3.

Now, we can compare both sides of the equation. We have 10^-3 = 10^-3. This is a true statement, which confirms that our solution, k = -1, is indeed correct. The verification process not only confirms the correctness of our solution but also reinforces our understanding of the exponential properties we used to solve the problem. It provides a sense of closure and confidence in our mathematical skills. Furthermore, the verification step is a valuable habit to cultivate in problem-solving. It helps identify errors, improve accuracy, and deepen understanding of the underlying concepts. In this case, the successful verification validates our step-by-step solution and solidifies our knowledge of exponential equations.

Alternative Verification Methods can sometimes provide additional assurance. For instance, we could convert the exponential terms to their decimal equivalents and perform the multiplication. 10^-3 is equal to 0.001, and 10^-1 is equal to 0.1. So, the left side of the equation becomes 0.001 * 10 * 0.1 = 0.001. This again confirms that our solution is correct. The use of multiple verification methods can be particularly helpful in more complex problems, where the chances of making an error are higher. In summary, the solution verification step is an indispensable part of the problem-solving process, ensuring accuracy, reinforcing understanding, and building confidence in our mathematical abilities.

Conclusion: Mastering Exponential Equations

In conclusion, our journey through solving the exponential equation 10^-3 ⋅ 10 ⋅ 10^k = 10^-3 has culminated in the determination that the value of k is -1. This process has not only provided a numerical answer but has also served as a valuable exercise in applying fundamental exponential properties and algebraic techniques. We began by understanding the problem statement, which involved isolating the unknown exponent k. We then strategically applied the product of powers property to simplify the equation, combining exponential terms with the same base. This simplification allowed us to transform the exponential equation into a more manageable algebraic equation.

Throughout the solution process, we emphasized the importance of understanding and applying exponential properties. These properties, such as the product of powers property and the negative exponent property, are the cornerstones of solving exponential equations. Without a firm grasp of these properties, manipulating and simplifying exponential expressions becomes a daunting task. By carefully applying these properties, we were able to reduce the complexity of the equation and isolate the unknown exponent k. This process highlights the power of mathematical tools and the importance of knowing when and how to apply them. The step-by-step approach we followed not only led us to the correct solution but also provided a clear and logical pathway that can be applied to other exponential equations.

The solution verification step further underscored the importance of accuracy and thoroughness in problem-solving. By substituting our solution, k = -1, back into the original equation, we were able to confirm that our answer was indeed correct. This verification process not only built confidence in our solution but also reinforced our understanding of the underlying concepts. It serves as a reminder that problem-solving is not just about finding an answer but also about ensuring that the answer is accurate and that the process used to arrive at the answer is sound. Mastering exponential equations requires a combination of conceptual understanding, strategic application of properties, and meticulous verification. The skills honed in this process are not only valuable in mathematics but also in various other fields that rely on exponential models, such as physics, computer science, and finance. Thus, the journey of solving for k in this equation is a testament to the power of mathematical reasoning and the importance of continuous learning and practice.

The value of k is -1, so the answer is B.

Keywords

exponential equations, product of powers property, negative exponent property, exponents, value of k