Solving The Exponential Equation 7 * 3^(x+1) - 5^(x+2) = 3^(x+4) - 5^(x-3)
Introduction to Exponential Equations
In the realm of mathematics, exponential equations hold a significant place, often appearing in various contexts such as calculus, algebra, and applied sciences. These equations involve variables in the exponents, presenting unique challenges and requiring specific techniques for their solution. This article delves into the process of solving the exponential equation 7 * 3^(x+1) - 5^(x+2) = 3^(x+4) - 5^(x-3), providing a step-by-step guide to understanding and tackling such problems. Exponential equations are not just abstract mathematical constructs; they model real-world phenomena like population growth, radioactive decay, and compound interest, making their understanding crucial. The equation at hand, 7 * 3^(x+1) - 5^(x+2) = 3^(x+4) - 5^(x-3), might seem daunting at first, but with the application of the right strategies and properties of exponents, it can be systematically solved. We'll break down the equation, manipulate its terms, and employ algebraic techniques to isolate the variable x, ultimately revealing the solution set. This exploration will not only solve this particular equation but also equip you with the tools to approach other exponential equations with confidence. Remember, the key to mastering exponential equations lies in understanding the fundamental properties of exponents and employing algebraic manipulations skillfully. So, let's embark on this mathematical journey and unravel the intricacies of this exponential equation, paving the way for a deeper understanding of exponential functions and their applications. Understanding exponential equations is a fundamental aspect of mathematical literacy, and this exercise serves as a valuable opportunity to hone your problem-solving skills and mathematical intuition. By carefully examining the structure of the equation and applying the appropriate techniques, we will transform it into a more manageable form, ultimately leading to the desired solution. This process highlights the elegance and power of mathematical reasoning, demonstrating how complex problems can be broken down into simpler, more approachable steps. The journey of solving this equation is as important as the solution itself, as it reinforces key mathematical concepts and cultivates a deeper appreciation for the interconnectedness of mathematical ideas. So, let's dive into the details and explore the world of exponential equations together.
Rewriting the Equation
To solve the given exponential equation 7 * 3^(x+1) - 5^(x+2) = 3^(x+4) - 5^(x-3), the initial step involves rewriting the equation to group like terms together. This means rearranging the terms so that terms with the same base are on the same side of the equation. The primary goal here is to consolidate the expressions involving the base 3 and the base 5, making it easier to identify common factors and simplify the equation. By strategically moving terms across the equals sign, we aim to create a more organized structure that facilitates further algebraic manipulation. This step is crucial because it sets the stage for applying the properties of exponents and logarithms, which are essential tools for solving exponential equations. The process of rearranging terms involves careful attention to the rules of algebraic manipulation, ensuring that the equality is maintained throughout the transformation. For instance, adding or subtracting the same quantity from both sides of the equation preserves the balance, while multiplying or dividing both sides by a non-zero quantity also maintains the equality. The ability to rearrange equations effectively is a fundamental skill in algebra, and it plays a vital role in solving a wide range of mathematical problems. In the context of exponential equations, this skill is particularly important because it allows us to isolate the exponential terms and prepare them for further simplification. The equation 7 * 3^(x+1) - 5^(x+2) = 3^(x+4) - 5^(x-3) can be rearranged by adding 5^(x-3) to both sides and subtracting 7 * 3^(x+1) from both sides. This process yields the equation - 5^(x+2) + 5^(x-3) = 3^(x+4) - 7 * 3^(x+1). This rearrangement is a key step in the solution process, as it brings together the terms with the same base, setting the stage for the next phase of simplification. It's a testament to the power of algebraic manipulation in transforming complex equations into more manageable forms. By understanding and applying these techniques, we can approach a wide variety of mathematical problems with confidence and clarity. This initial rearrangement is not just a mechanical step; it's a strategic move that lays the foundation for the subsequent steps in the solution process. It demonstrates the importance of careful planning and organization in problem-solving, highlighting the value of breaking down complex problems into smaller, more manageable components.
Applying Exponent Rules
After rewriting the equation - 5^(x+2) + 5^(x-3) = 3^(x+4) - 7 * 3^(x+1), the next critical step involves applying the rules of exponents to further simplify the equation. Exponent rules provide a powerful set of tools for manipulating expressions with exponents, allowing us to combine terms, factor out common factors, and ultimately isolate the variable x. One of the most fundamental exponent rules states that a^(m+n) = a^m * a^n. This rule allows us to break down exponents that involve sums or differences into products or quotients of exponents. Another important rule is a^(m-n) = a^m / a^n, which is particularly useful for dealing with exponents that involve subtraction. By applying these rules strategically, we can transform the exponential terms in the equation into a more manageable form, paving the way for further simplification and solution. The application of exponent rules is not just a mechanical process; it requires a deep understanding of the properties of exponents and a keen eye for identifying opportunities for simplification. Each term in the equation must be carefully examined to determine how exponent rules can be applied to make it more amenable to algebraic manipulation. For instance, the term 5^(x+2) can be rewritten as 5^x * 5^2, while the term 5^(x-3) can be rewritten as 5^x / 5^3. Similarly, the term 3^(x+4) can be rewritten as 3^x * 3^4, and the term 3^(x+1) can be rewritten as 3^x * 3^1. By applying these transformations, we can create opportunities to factor out common factors and simplify the equation. The strategic use of exponent rules is a key skill in solving exponential equations, and it demonstrates the power of mathematical notation in capturing complex relationships in a concise and elegant manner. This step is not just about simplifying the equation; it's about gaining a deeper understanding of the underlying mathematical structure and leveraging that understanding to solve the problem. The careful application of exponent rules transforms the equation from a seemingly complex expression into a more manageable form, highlighting the importance of mathematical tools in problem-solving. This process underscores the beauty and elegance of mathematics, where seemingly disparate concepts come together to provide a solution.
Factoring and Simplifying
Following the application of exponent rules, the equation - 5^(x+2) + 5^(x-3) = 3^(x+4) - 7 * 3^(x+1) transforms into -5^x * 5^2 + 5^x / 5^3 = 3^x * 3^4 - 7 * 3^x * 3^1. This form of the equation presents a clear opportunity for factoring. Factoring is a fundamental algebraic technique that involves identifying common factors in an expression and extracting them to simplify the expression. In this case, the common factor on the left side of the equation is 5^x, and the common factor on the right side is 3^x. By factoring out these common factors, we can significantly simplify the equation and make it more amenable to further manipulation. Factoring is not just a mechanical process; it requires a keen eye for patterns and a deep understanding of algebraic structure. It's a powerful tool that allows us to break down complex expressions into simpler, more manageable components. The ability to factor effectively is a crucial skill in algebra, and it plays a vital role in solving a wide range of mathematical problems. The process of factoring involves carefully examining the terms in an expression and identifying the common factors that can be extracted. Once the common factors have been identified, they can be factored out, leaving behind a simplified expression. In the context of this equation, factoring out 5^x from the left side gives us 5^x (-5^2 + 1/5^3), and factoring out 3^x from the right side gives us 3^x (3^4 - 7 * 3). This factoring step is a crucial milestone in the solution process, as it allows us to isolate the exponential terms and prepare them for further simplification. The factored equation, 5^x (-5^2 + 1/5^3) = 3^x (3^4 - 7 * 3), can be further simplified by evaluating the constants within the parentheses. This simplification involves performing basic arithmetic operations, such as squaring, cubing, and multiplying. The goal is to reduce the constants to their simplest form, making the equation even more manageable. By simplifying the constants, we can gain a clearer understanding of the relationship between the exponential terms and the constant terms in the equation. This simplification process is not just about making the equation look simpler; it's about gaining a deeper understanding of the underlying mathematical structure. The simplified equation sets the stage for the next crucial step in the solution process: isolating the exponential terms and solving for x. This process underscores the importance of careful attention to detail and the power of algebraic manipulation in solving mathematical problems.
Isolating Exponential Terms
After factoring and simplifying, our equation stands as 5^x (-5^2 + 1/5^3) = 3^x (3^4 - 7 * 3). The next strategic step is to isolate the exponential terms. This involves manipulating the equation to get the terms with x on one side and the constant terms on the other. This isolation is crucial because it allows us to directly compare the exponential expressions and ultimately solve for x. The most common technique for isolating exponential terms in this type of equation is to divide both sides by one of the exponential terms, effectively creating a ratio of exponential expressions. This step is not just a mechanical manipulation; it's a strategic move that leverages the properties of exponents and fractions to simplify the equation. By creating a ratio of exponential expressions, we can often reveal a pattern or relationship that allows us to solve for x. The process of isolating exponential terms requires careful attention to the rules of algebra, ensuring that the equality is maintained throughout the transformation. For instance, dividing both sides of the equation by the same non-zero quantity preserves the balance, while multiplying both sides by the same quantity also maintains the equality. The ability to isolate variables and terms is a fundamental skill in algebra, and it plays a vital role in solving a wide range of mathematical problems. In the context of exponential equations, this skill is particularly important because it allows us to focus on the exponential expressions and apply the appropriate techniques to solve for x. To isolate the exponential terms in our equation, we can divide both sides by 3^x and by (-5^2 + 1/5^3). This yields the equation 5^x / 3^x = (3^4 - 7 * 3) / (-5^2 + 1/5^3). This isolation step is a key milestone in the solution process, as it brings together the exponential terms on one side of the equation, setting the stage for the next phase of simplification. It's a testament to the power of algebraic manipulation in transforming complex equations into more manageable forms. By understanding and applying these techniques, we can approach a wide variety of mathematical problems with confidence and clarity. This isolation is not just a mechanical step; it's a strategic move that lays the foundation for the subsequent steps in the solution process. It demonstrates the importance of careful planning and organization in problem-solving, highlighting the value of breaking down complex problems into smaller, more manageable components.
Applying Logarithms
Having isolated the exponential terms as 5^x / 3^x = (3^4 - 7 * 3) / (-5^2 + 1/5^3), we now encounter a situation where logarithms become indispensable. Logarithms are the inverse operation to exponentiation, and they provide a powerful tool for solving equations where the variable is in the exponent. The key property of logarithms that makes them so useful in this context is the power rule, which states that log_b(a^c) = c * log_b(a). This rule allows us to bring the exponent down as a coefficient, effectively transforming an exponential equation into a linear equation that can be easily solved. The choice of logarithm base is often arbitrary, but common choices include base 10 (common logarithm) and base e (natural logarithm, denoted as ln). The natural logarithm is particularly useful in calculus and other advanced mathematical contexts, but for the purpose of solving this equation, either base 10 or base e will work equally well. The application of logarithms is not just a mechanical process; it requires a deep understanding of the properties of logarithms and a keen eye for identifying opportunities to simplify the equation. Each term in the equation must be carefully examined to determine how logarithms can be applied to make it more amenable to algebraic manipulation. In our case, we can apply the logarithm to both sides of the equation, which maintains the equality and allows us to utilize the power rule. The application of logarithms is a key skill in solving exponential equations, and it demonstrates the power of mathematical notation in capturing complex relationships in a concise and elegant manner. This step is not just about simplifying the equation; it's about gaining a deeper understanding of the underlying mathematical structure and leveraging that understanding to solve the problem. To apply logarithms to our equation, we can take the natural logarithm (ln) of both sides, resulting in ln(5^x / 3^x) = ln((3^4 - 7 * 3) / (-5^2 + 1/5^3)). This step sets the stage for the next crucial step in the solution process: applying the properties of logarithms to further simplify the equation and isolate x. The careful application of logarithms transforms the equation from a seemingly complex expression into a more manageable form, highlighting the importance of mathematical tools in problem-solving. This process underscores the beauty and elegance of mathematics, where seemingly disparate concepts come together to provide a solution.
Solving for x
After applying logarithms, our equation transforms into ln(5^x / 3^x) = ln((3^4 - 7 * 3) / (-5^2 + 1/5^3)). To proceed, we can use the logarithm properties to simplify further. Specifically, we can use the quotient rule, which states that ln(a/b) = ln(a) - ln(b), and the power rule, which states that ln(a^c) = c * ln(a). Applying these rules to the left side of the equation gives us ln(5^x) - ln(3^x) = x * ln(5) - x * ln(3). On the right side, we have ln((3^4 - 7 * 3) / (-5^2 + 1/5^3)), which is a constant value that can be calculated. Now, our equation looks like x * ln(5) - x * ln(3) = ln((3^4 - 7 * 3) / (-5^2 + 1/5^3)). We can factor out x from the left side, giving us x * (ln(5) - ln(3)) = ln((3^4 - 7 * 3) / (-5^2 + 1/5^3)). To finally solve for x, we divide both sides by (ln(5) - ln(3)), resulting in x = ln((3^4 - 7 * 3) / (-5^2 + 1/5^3)) / (ln(5) - ln(3)). This is the exact solution for x. To obtain a numerical approximation, we can use a calculator to evaluate the logarithmic expressions and perform the division. This process involves using the calculator's natural logarithm function (usually labeled as "ln") to compute the logarithms of the various numerical values in the equation. The numerical approximation provides a concrete value for x, which can be useful for understanding the magnitude of the solution and for verifying the solution in the original equation. The process of solving for x using logarithms highlights the power of mathematical tools in transforming complex equations into manageable forms. By carefully applying the properties of logarithms and algebraic manipulation, we can isolate the variable and obtain a solution. This process underscores the importance of a solid foundation in mathematical concepts and techniques, as well as the ability to apply those concepts in a strategic and effective manner. The final solution, whether expressed in exact form or as a numerical approximation, represents the culmination of a step-by-step problem-solving process that demonstrates the elegance and power of mathematical reasoning.
Conclusion
In conclusion, solving the exponential equation 7 * 3^(x+1) - 5^(x+2) = 3^(x+4) - 5^(x-3) involved a multi-faceted approach, showcasing the elegance and power of mathematical problem-solving. We began by rewriting the equation to group like terms, setting the stage for simplification. Then, we skillfully applied exponent rules to break down the exponential terms into more manageable components. Factoring out common factors allowed us to further simplify the equation, revealing the underlying structure and paving the way for isolating the variable x. The strategic application of logarithms provided the crucial tool for dealing with the variable in the exponent, transforming the equation into a form that could be readily solved. Finally, through careful algebraic manipulation, we isolated x and obtained both the exact solution and a numerical approximation. This journey through the solution process highlights the importance of a solid foundation in mathematical concepts, including exponent rules, factoring techniques, and the properties of logarithms. It also underscores the value of a systematic and organized approach to problem-solving, where each step builds upon the previous one to lead to the desired solution. The ability to solve exponential equations is not just a theoretical exercise; it has practical applications in various fields, such as finance, physics, and engineering, where exponential models are used to describe phenomena like compound interest, radioactive decay, and population growth. By mastering the techniques for solving exponential equations, we equip ourselves with a valuable tool for understanding and analyzing the world around us. The process of solving this equation also demonstrates the interconnectedness of mathematical ideas, where concepts from algebra, calculus, and trigonometry often come together to provide a solution. This interconnectedness is a hallmark of mathematics, and it highlights the importance of developing a holistic understanding of the subject. The solution to this exponential equation is not just a numerical value; it's a testament to the power of mathematical reasoning and the beauty of mathematical problem-solving. It's a reminder that even complex problems can be tackled with the right tools and the right approach. By embracing the challenges presented by exponential equations and other mathematical problems, we can cultivate our problem-solving skills and deepen our appreciation for the elegance and power of mathematics.
