Solving Exponential Equations A Step-by-Step Guide To 2^(x+1)/(x+4) = 2^(x-1)/(x+2)
Introduction to Exponential Equations
In the realm of mathematics, exponential equations present a fascinating challenge, combining the elegance of algebra with the unique properties of exponential functions. These equations, characterized by the presence of a variable in the exponent, often require a blend of algebraic manipulation and a deep understanding of exponential rules to solve. This article delves into the intricacies of solving one such equation: 2^(x+1)/(x+4) = 2^(x-1)/(x+2). We will explore the step-by-step process, highlighting the key mathematical principles and techniques involved. Understanding how to solve exponential equations is crucial not only for academic pursuits but also for various real-world applications, such as modeling population growth, radioactive decay, and compound interest calculations.
Understanding the Basics of Exponential Functions
Before diving into the solution, let's briefly revisit the fundamentals of exponential functions. An exponential function is a function of the form f(x) = a^x, where 'a' is a constant called the base, and 'x' is the exponent. The base 'a' is typically a positive real number not equal to 1. The exponent 'x' can be any real number. The key property of exponential functions is that the function value changes rapidly as the exponent changes, leading to exponential growth or decay depending on whether the base 'a' is greater than 1 or between 0 and 1, respectively. In our equation, the base is 2, which signifies exponential growth. Understanding the behavior of exponential functions, such as their rapid growth and decay, is essential for solving exponential equations effectively. Furthermore, familiarity with the laws of exponents, such as the product rule, quotient rule, and power rule, is indispensable for simplifying and manipulating exponential expressions. The ability to express numbers in exponential form and to convert between exponential and logarithmic forms also plays a vital role in solving these equations.
Rewriting the Equation for Clarity
The given equation is 2^(x+1)/(x+4) = 2^(x-1)/(x+2). To approach this problem effectively, the first step involves rewriting the equation in a more manageable form. This typically entails eliminating fractions and rearranging terms to isolate the exponential expressions. By cross-multiplying, we can eliminate the denominators and obtain an equivalent equation without fractions. This step is crucial because it simplifies the equation and makes it easier to apply further algebraic manipulations. The rewritten equation will then allow us to leverage the properties of exponents more readily. Cross-multiplication is a fundamental technique in algebra that transforms an equation involving fractions into a more straightforward linear or polynomial equation, which is often easier to solve. In the context of exponential equations, cross-multiplication helps to consolidate the exponential terms, making it possible to apply the laws of exponents effectively. Moreover, this step underscores the importance of understanding and applying basic algebraic principles in conjunction with exponential concepts.
Step-by-Step Solution
1. Cross-Multiplication
To begin, we cross-multiply to eliminate the fractions: 2^(x+1) * (x+2) = 2^(x-1) * (x+4). This step is critical as it transforms the equation into a more manageable form, paving the way for further simplification. Cross-multiplication is a fundamental technique in algebra used to eliminate fractions from an equation. By multiplying the numerator of the left side by the denominator of the right side and vice versa, we create a new equation that is free of fractions. This not only simplifies the equation but also makes it easier to apply other algebraic techniques, such as combining like terms and factoring. In the context of exponential equations, cross-multiplication is particularly useful because it allows us to consolidate the exponential terms and prepare the equation for the application of exponential properties. The resulting equation after cross-multiplication sets the stage for the next steps in the solution process, where we will leverage the laws of exponents to further simplify and solve for the unknown variable.
2. Applying Exponential Properties
Next, we apply the properties of exponents to simplify the equation. Specifically, we can rewrite 2^(x+1) as 2^x * 2^1 and 2^(x-1) as 2^x * 2^(-1). Substituting these expressions back into the equation, we get: (2^x * 2) * (x+2) = (2^x * 2^(-1)) * (x+4). This step leverages the fundamental property of exponents that states a^(m+n) = a^m * a^n. By applying this property, we can break down the exponential terms into simpler components, which facilitates the process of solving the equation. The ability to manipulate exponential expressions using the laws of exponents is a crucial skill in solving exponential equations. It allows us to rewrite complex expressions in a more manageable form, making it easier to isolate the variable and find its value. In this case, breaking down the exponential terms allows us to identify a common factor, which we will eliminate in the next step, further simplifying the equation. This step highlights the interconnectedness of algebraic principles and exponential properties in solving mathematical problems.
3. Simplifying and Cancelling Common Factors
Observe that 2^x is a common factor on both sides of the equation. Dividing both sides by 2^x, we get: 2 * (x+2) = 2^(-1) * (x+4). This simplification is a crucial step, as it significantly reduces the complexity of the equation. Dividing both sides of an equation by a common factor is a fundamental algebraic technique used to simplify equations and make them easier to solve. In this case, the common factor is the exponential term 2^x, which appears on both sides of the equation. By dividing both sides by this factor, we eliminate it from the equation, leaving us with a simpler expression that involves only linear terms. This simplification is particularly effective in exponential equations because it allows us to isolate the variable from the exponential terms, making it easier to apply standard algebraic techniques to solve for the variable. This step demonstrates the power of algebraic manipulation in simplifying complex equations and paving the way for a straightforward solution.
4. Solving the Linear Equation
Now we have a linear equation: 2(x+2) = (1/2)(x+4). Expanding and simplifying, we get: 2x + 4 = (1/2)x + 2. Multiplying both sides by 2 to eliminate the fraction, we have: 4x + 8 = x + 4. Subtracting x and 8 from both sides, we get: 3x = -4. Finally, dividing by 3, we find: x = -4/3. Solving linear equations is a fundamental skill in algebra, and it is often a necessary step in solving more complex equations, including exponential equations. Linear equations are characterized by the fact that the variable appears only to the first power, and they can be solved using a series of algebraic manipulations, such as adding or subtracting the same quantity from both sides, multiplying or dividing both sides by the same non-zero quantity, and combining like terms. In this case, we start by expanding the equation to remove the parentheses and then apply a series of algebraic operations to isolate the variable x on one side of the equation. The resulting value of x is the solution to the linear equation and also the solution to the original exponential equation. This step demonstrates the importance of mastering basic algebraic skills in order to tackle more advanced mathematical problems.
5. Verification of the Solution
It is crucial to verify the solution by substituting x = -4/3 back into the original equation. This step ensures that our solution is valid and does not introduce any extraneous roots. Substituting x = -4/3 into the original equation, we have: 2^(-4/3 + 1) / (-4/3 + 4) = 2^(-4/3 - 1) / (-4/3 + 2). Simplifying the exponents and the denominators, we get: 2^(-1/3) / (8/3) = 2^(-7/3) / (2/3). Further simplification shows that both sides of the equation are equal, confirming that x = -4/3 is indeed the correct solution. Verification is a critical step in solving any equation, especially exponential equations, because it helps to ensure that the solution obtained is valid and does not lead to any contradictions or inconsistencies. In some cases, the algebraic manipulations performed to solve an equation may introduce extraneous roots, which are solutions that do not satisfy the original equation. By substituting the solution back into the original equation, we can check whether it is a true solution or an extraneous root. This step provides a crucial check on the accuracy of our solution and helps to avoid errors. In this case, the verification step confirms that x = -4/3 is a valid solution to the original exponential equation.
Conclusion
In conclusion, the solution to the equation 2^(x+1)/(x+4) = 2^(x-1)/(x+2) is x = -4/3. This problem showcases the importance of understanding and applying exponential properties, as well as basic algebraic techniques. Solving exponential equations often requires a multi-step approach, involving simplification, manipulation, and verification. The ability to solve exponential equations is a valuable skill in mathematics, with applications in various fields of science and engineering. By mastering the techniques demonstrated in this article, students and practitioners can confidently tackle a wide range of exponential equation problems. The solution process not only provides the answer to a specific problem but also reinforces the understanding of fundamental mathematical principles and problem-solving strategies. Furthermore, the verification step underscores the importance of accuracy and attention to detail in mathematical calculations. The successful solution of this equation exemplifies the power of combining algebraic and exponential concepts to solve challenging mathematical problems.
