Solving For K In Exponential Equations A Step-by-Step Guide

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In the realm of mathematics, exponential equations play a crucial role in various fields, ranging from finance to physics. Understanding how to solve these equations is essential for anyone seeking to master mathematical concepts. This article delves into the process of finding the value of K in two distinct exponential equations. We will break down each equation step-by-step, providing clear explanations and insights into the underlying principles. Whether you are a student grappling with exponents or a seasoned mathematician looking for a refresher, this guide will equip you with the knowledge and skills to tackle similar problems with confidence. We aim to provide a comprehensive understanding of exponential equations, empowering you to solve them effectively and efficiently. This exploration will not only enhance your problem-solving abilities but also deepen your appreciation for the elegance and power of mathematics.

Breaking Down the Equation

To solve the equation 125 × 5^6 = 5^K, our primary goal is to express both sides of the equation with the same base. This is a fundamental strategy in solving exponential equations, as it allows us to equate the exponents directly. Recognizing that 125 is a power of 5 is the first key step. Specifically, 125 can be written as 5^3. Substituting this into our equation, we get 5^3 × 5^6 = 5^K. Now, we have a product of powers with the same base on the left side. The rule for multiplying powers with the same base states that we add the exponents. Applying this rule, we have 5^(3+6) = 5^K, which simplifies to 5^9 = 5^K. With both sides now expressed as powers of 5, we can directly equate the exponents. This gives us K = 9. Therefore, the value of K that satisfies the equation is 9. This process demonstrates a core principle in solving exponential equations: simplifying expressions to a common base to equate exponents. This approach is applicable to a wide range of problems and is a cornerstone of exponential equation manipulation. By mastering this technique, you can effectively solve more complex problems and deepen your understanding of exponential functions.

Step-by-Step Solution

  1. Express 125 as a power of 5: The first crucial step is recognizing that 125 can be expressed as 5 raised to the power of 3. This is because 5 × 5 × 5 = 125, which can be written as 5^3. This transformation is essential for aligning the bases on both sides of the equation, a critical step in solving exponential equations.
  2. Substitute 5^3 into the equation: Replacing 125 with 5^3 in the original equation 125 × 5^6 = 5^K gives us 5^3 × 5^6 = 5^K. This substitution is a key maneuver, bringing us closer to having a common base on both sides of the equation.
  3. Apply the product of powers rule: The product of powers rule states that when multiplying powers with the same base, you add the exponents. In this case, we have 5^3 × 5^6, which simplifies to 5^(3+6). This is a fundamental rule in exponent manipulation and is crucial for simplifying expressions.
  4. Simplify the exponent: Adding the exponents 3 and 6 gives us 9. So, the equation becomes 5^9 = 5^K. This simplification is a direct application of the product of powers rule and brings the equation to a more manageable form.
  5. Equate the exponents: Now that both sides of the equation have the same base (5), we can equate the exponents. This means that if 5^9 = 5^K, then 9 = K. This is a critical step in solving exponential equations where the bases are the same. By equating the exponents, we directly find the value of the unknown variable.
  6. State the solution: Therefore, the value of K is 9. This is the final answer, the solution to the original exponential equation. This step-by-step solution demonstrates the methodical approach required to solve exponential equations, emphasizing the importance of simplifying expressions and aligning bases.

Solving the Second Equation

In this equation, 3^K × 9 = 3^13}*, our objective remains the same to express both sides with a common base. We recognize that 9 is a power of 3, specifically 3^2. Substituting this into the equation gives us *3^K × 3^2 = 3^{13. Applying the product of powers rule, we add the exponents on the left side, resulting in 3^(K+2) = 3^{13}. Now, with both sides expressed as powers of 3, we can equate the exponents. This leads to the equation K + 2 = 13. To solve for K, we subtract 2 from both sides, yielding K = 11. Thus, the value of K that satisfies the equation is 11. This problem reinforces the importance of recognizing common bases and applying exponent rules to simplify and solve exponential equations. The process of identifying the base, applying the product of powers rule, and equating exponents is a fundamental strategy in handling these types of problems. By mastering this approach, you can confidently tackle a wide range of exponential equations, enhancing your mathematical problem-solving skills.

Detailed Solution Steps

  1. Express 9 as a power of 3: The first step in solving this equation is to recognize that 9 can be expressed as 3 squared, or 3^2. This is a crucial step in unifying the bases on both sides of the equation.
  2. Substitute 3^2 into the equation: Replacing 9 with 3^2 in the original equation 3^K × 9 = 3^{13} gives us 3^K × 3^2 = 3^{13}. This substitution is vital for applying the rules of exponents and simplifying the equation.
  3. Apply the product of powers rule: The product of powers rule dictates that when multiplying powers with the same base, you add the exponents. Applying this rule to the left side of the equation, 3^K × 3^2 becomes 3^(K+2). This rule is fundamental in exponent manipulation and is essential for solving exponential equations.
  4. Rewrite the equation: After applying the product of powers rule, the equation is rewritten as 3^(K+2) = 3^{13}. This form of the equation highlights the common base and sets the stage for equating the exponents.
  5. Equate the exponents: With the same base on both sides, we can equate the exponents. This means that K + 2 = 13. This is a pivotal step in isolating the unknown variable, K.
  6. Solve for K: To solve for K, we subtract 2 from both sides of the equation K + 2 = 13. This gives us K = 13 - 2, which simplifies to K = 11. This algebraic manipulation isolates K and provides its value.
  7. State the solution: Therefore, the value of K is 11. This is the final answer to the equation. This detailed solution illustrates the systematic approach to solving exponential equations, emphasizing the importance of base unification and exponent manipulation.

In summary, we've successfully navigated through two exponential equations, demonstrating the fundamental techniques required to find the value of K. These techniques include expressing numbers as powers of a common base, applying the product of powers rule, and equating exponents. Mastering these skills is essential for tackling more complex mathematical problems. The ability to manipulate exponential equations is not just a mathematical skill; it's a valuable tool in various fields, from science and engineering to finance and computer science. By understanding the underlying principles and practicing these techniques, you can build a strong foundation in mathematics and enhance your problem-solving abilities. The journey through these equations highlights the elegance and power of mathematical reasoning, empowering you to approach future challenges with confidence and competence. Whether you're a student preparing for exams or a professional applying mathematical concepts in your field, the knowledge gained here will serve as a valuable asset in your intellectual toolkit. Remember, mathematics is not just about formulas and equations; it's about understanding the logic and patterns that govern the world around us.