Solving Exponential Equations A Step-by-Step Guide For 27=(1/3)^(x/2)

Introduction

In this article, we will dive into solving an exponential equation. Exponential equations are equations where the variable appears in the exponent. These types of equations are common in various fields, including mathematics, physics, engineering, and finance. The equation we aim to solve is $27 = \left(\frac{1}{3}\right)^{\frac{x}{2}}$. This equation might seem daunting at first, but with a systematic approach and the application of logarithmic properties, we can find the value of $x$ that satisfies it. Understanding the properties of exponents and logarithms is crucial in solving these equations. Exponential equations form the basis for many mathematical models, describing phenomena such as population growth, radioactive decay, and compound interest. By mastering the techniques to solve them, we unlock the ability to analyze and predict these real-world processes. The solution involves converting the equation into a form where both sides have the same base, which then allows us to equate the exponents. This article will guide you through each step, ensuring a clear understanding of the process. We'll explore the fundamental principles behind exponential equations, their characteristics, and the methods used to tackle them effectively. So, let's embark on this mathematical journey together and unravel the mystery behind this equation.

Understanding Exponential Equations

Before we jump into the solution, let's first understand what an exponential equation is and its general form. An exponential equation is an equation in which the variable occurs in the exponent. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result. In our case, the equation $27 = \left(\frac{1}{3}\right)^{\frac{x}{2}}$ fits this form, where the base is $\frac{1}{3}$, the exponent is $\frac{x}{2}$, and the result is $27$. Exponential functions are characterized by their rapid growth or decay, depending on whether the base is greater than 1 or between 0 and 1, respectively. Understanding the base is crucial in solving these equations because it allows us to rewrite the equation in a more manageable form. In this particular problem, we see that the base on one side is a whole number (27), while the base on the other side is a fraction (1/3). This difference highlights the need for a common base to simplify the equation and make it solvable. Furthermore, the exponent ${\frac{x}{2}}$ introduces another layer of complexity, as the variable is divided by 2. This fractional exponent indicates that the variable's effect on the equation's solution may be tempered, requiring careful consideration when manipulating the equation. As we proceed, we'll see how converting the bases to a common number and properly handling the exponent will lead us to the solution. Understanding exponential equations is not just about manipulating numbers; it's about grasping the fundamental relationships that govern growth and decay in various natural and artificial systems.

Rewriting the Equation with a Common Base

The key to solving this exponential equation is to rewrite both sides with the same base. We notice that $27$ can be expressed as $3^3$, and $\frac{1}{3}$ can be written as $3^{-1}$. Thus, we can rewrite the equation as $3^3 = (3^{-1})^{\frac{x}{2}}$. This step is crucial because it allows us to directly compare the exponents once we simplify further. The process of finding a common base is fundamental in solving exponential equations because it transforms the equation into a form where the exponents can be equated. This simplification is based on the property that if $a^m = a^n$, then $m = n$, provided that $a$ is a positive number not equal to 1. In our case, we are choosing the common base of 3, which allows us to express both sides of the equation in terms of powers of 3. The ability to manipulate numbers and recognize these relationships is a cornerstone of mathematical problem-solving. Furthermore, the conversion of $\frac{1}{3}$ to $3^{-1}$ demonstrates an important property of exponents: the reciprocal of a number raised to a power is equal to the number raised to the negative of that power. This property is frequently used in simplifying exponential expressions and equations. As we continue, we will simplify the right-hand side of the equation, making use of the power of a power rule, which states that $(a^m)^n = a^{mn}$. This rule will enable us to further consolidate the terms and bring us closer to isolating the variable $x$.

Simplifying the Exponents

Now that we have $3^3 = (3^{-1})^{\frac{x}{2}}$, we can use the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule to the right side of the equation, we get $3^3 = 3^{-\frac{x}{2}}$. This step significantly simplifies the equation by eliminating the parentheses and combining the exponents. The power of a power rule is a fundamental concept in exponent manipulation and is essential for simplifying complex exponential expressions. It essentially states that when you raise a power to another power, you multiply the exponents. This rule is widely applicable in various mathematical contexts, including algebra, calculus, and even more advanced fields like differential equations. In our specific problem, this rule allows us to transform the right-hand side of the equation into a form that is directly comparable to the left-hand side. The exponent $-\frac{x}{2}$ now represents the combined effect of raising $\frac{1}{3}$ to the power of $\frac{x}{2}$. This negative exponent also indicates that as $x$ increases, the value of $3^{-\frac{x}{2}}$ decreases, reflecting an inverse relationship. By simplifying the exponents, we have brought the equation to a state where the bases are equal, and the exponents can be directly compared. This sets the stage for the next step, where we will equate the exponents and solve for the unknown variable $x$. This process highlights the elegance and efficiency of mathematical techniques in simplifying complex problems into manageable forms.

Equating the Exponents and Solving for x

Since the bases are now the same, we can equate the exponents. This gives us the equation $3 = -\frac{x}{2}$. To solve for $x$, we multiply both sides of the equation by $-2$, resulting in $x = -6$. Therefore, the solution to the equation $27 = \left(\frac{1}{3}\right)^{\frac{x}{2}}$ is $x = -6$. This step is the culmination of our previous efforts, where we strategically manipulated the equation to isolate the variable. Equating the exponents is a direct consequence of the fundamental property of exponential functions: if $a^m = a^n$, then $m = n$, provided that $a$ is a positive number not equal to 1. This property is the cornerstone of solving exponential equations and allows us to transform a complex equation into a simple algebraic one. The resulting equation, $3 = -\frac{x}{2}$, is a linear equation that can be easily solved using basic algebraic principles. Multiplying both sides by -2 is a standard algebraic technique to eliminate the fraction and isolate $x$. This operation maintains the equality of the equation, ensuring that the solution remains valid. The solution, $x = -6$, is a single numerical value that satisfies the original exponential equation. This value can be verified by substituting it back into the original equation and confirming that both sides are equal. The negative sign indicates that the exponent is acting in a way that inverts the base, which aligns with the fact that $\frac{1}{3}$ is the reciprocal of 3. The process of solving for $x$ demonstrates the power of mathematical reasoning in unraveling complex relationships and arriving at precise solutions. By meticulously following the steps and applying the correct principles, we have successfully solved the exponential equation.

Verification of the Solution

To ensure the accuracy of our solution, it's essential to verify it by substituting $x = -6$ back into the original equation: $27 = \left(\frac{1}{3}\right)^{\frac{x}{2}}$. Substituting $x = -6$, we get $27 = \left(\frac{1}{3}\right)^{\frac{-6}{2}}$, which simplifies to $27 = \left(\frac{1}{3}\right)^{-3}$. Since $\left(\frac{1}{3}\right)^{-3} = 3^3 = 27$, our solution is indeed correct. Verification is a crucial step in the problem-solving process, especially in mathematics, as it confirms that the derived solution satisfies the original conditions of the problem. This step adds a layer of confidence in the correctness of the answer and helps identify any potential errors in the solution process. By substituting the value of $x$ back into the original equation, we are essentially testing whether the left-hand side and the right-hand side of the equation are equal. If they are, then the solution is verified; if not, then there is an error somewhere in the solution process that needs to be investigated. In our case, substituting $x = -6$ into the equation leads to a clear confirmation that both sides are equal, thus validating our solution. The verification process also serves as a useful exercise in reinforcing the concepts and techniques used in solving the problem. It allows us to revisit the steps we took and ensure that each step was logically sound and mathematically accurate. Moreover, verification can sometimes reveal alternative approaches to solving the problem, leading to a deeper understanding of the underlying principles. In summary, the verification step is an indispensable part of problem-solving and should always be performed to ensure the accuracy and reliability of the results.

Conclusion

In conclusion, we have successfully solved the exponential equation $27 = \left(\frac{1}{3}\right)^{\frac{x}{2}}$ by rewriting both sides with a common base, simplifying the exponents, equating the exponents, and solving for $x$. The solution we found is $x = -6$, which we verified by substituting it back into the original equation. This problem illustrates the importance of understanding the properties of exponents and the techniques for manipulating exponential equations. Solving exponential equations is a fundamental skill in mathematics, with applications in various fields. By mastering this skill, one can tackle a wide range of problems involving growth, decay, and other exponential phenomena. The process we followed highlights the systematic approach required to solve mathematical problems, which includes understanding the problem, devising a plan, executing the plan, and reviewing the solution. Each step in the solution process, from finding a common base to equating exponents, is based on well-established mathematical principles. By applying these principles correctly, we were able to transform a seemingly complex equation into a simple algebraic one. Furthermore, the verification step reinforces the importance of ensuring the accuracy of the solution. The ability to solve exponential equations is not just a matter of following a set of rules; it also involves developing a deeper understanding of the underlying mathematical concepts. This understanding enables one to adapt to different problem scenarios and apply the techniques creatively and effectively. As we continue to explore mathematical concepts, the skills and knowledge we have gained from solving this equation will serve as a valuable foundation for tackling more advanced problems.