Solving Exponential Equations Finding The Equivalent Of (1/3)^x = 27^(x+2)

Jul 11, 2025 by Admin 75 views

Unlocking the Equivalent Equation of (1/3)^x = 27^(x+2)

Introduction

In the realm of mathematics, particularly in the study of exponential equations, a common task involves finding equivalent forms of a given equation. This often simplifies the process of solving for the unknown variable. In this article, we delve into the equation (1/3)^x = 27^(x+2), dissecting it step by step to identify its equivalent form. Understanding exponential equations and the rules governing exponents is crucial for success in algebra and calculus. This exploration will not only provide the answer but also illuminate the underlying principles, making it easier to tackle similar problems in the future. The ability to manipulate exponents and express numbers in different bases is a fundamental skill that opens doors to more advanced mathematical concepts.

Problem Statement

The initial problem presents us with the equation:

(1/3)^x = 27^(x+2)

Our objective is to determine which of the provided options is equivalent to this equation. To achieve this, we'll leverage the properties of exponents and logarithms, transforming the equation into a more manageable form. The process involves expressing both sides of the equation with the same base, which allows us to equate the exponents. This technique is a cornerstone in solving exponential equations and is widely applicable across various mathematical contexts. By carefully applying the rules of exponents, we can systematically reduce the complexity of the equation and arrive at its equivalent form.

Step-by-Step Solution

Express both sides with the same base:

The key to solving this equation lies in recognizing that both 1/3 and 27 can be expressed as powers of 3. Specifically, 1/3 is 3^(-1) and 27 is 3^3. Rewriting the equation using this common base is the first critical step in simplifying the problem.

(1/3)^x = (3^(-1))x

27^(x+2) = (3³⁾(x+2)

This transformation allows us to apply the power of a power rule, which states that (a^m)n = a^(m*n). By expressing both sides of the equation with the same base, we set the stage for equating the exponents, a technique that is fundamental to solving exponential equations. This step is not just a mathematical manipulation; it is a strategic move that simplifies the problem and makes it solvable.

Apply the power of a power rule:

Now, we apply the power of a power rule, which states that (a^m)n = a^(m*n). This rule is essential for simplifying expressions where an exponent is raised to another exponent. Applying this rule to both sides of the equation, we get:

(3^(-1))x = 3^(-x)

(3³⁾(x+2) = 3^(3(x+2)) = 3^(3x+6)

By applying the power of a power rule, we have successfully simplified both sides of the equation, making it easier to compare the exponents. This step is crucial for reducing the complexity of the equation and revealing its underlying structure. The ability to apply this rule correctly is a testament to a solid understanding of exponent properties.

Set the exponents equal:

With both sides of the equation now expressed as powers of the same base (3), we can equate the exponents. This is a direct consequence of the property that if a^m = a^n, then m = n. Equating the exponents, we get:

-x = 3x + 6

This equation is a linear equation in x, which is much simpler to solve than the original exponential equation. This step highlights the power of transforming equations into equivalent forms that are easier to handle. The ability to recognize and apply this property is a key skill in solving exponential equations.

The equivalent equation:

The equation we derived by equating the exponents is:

3^(-x) = 3^(3x+6)

This equation directly corresponds to one of the given options, making it the equivalent form of the original equation. This final step confirms our solution and demonstrates the effectiveness of our step-by-step approach. By carefully applying the rules of exponents and simplifying the equation, we have successfully identified its equivalent form.

Analyzing the Options

Now, let's examine the provided options in light of our solution:

A. 3^x = 3^(-3x+2)
B. 3^x = 3^(3x+6)
C. 3^(-x) = 3^(3x+6)
D. -x = 3x+6

Comparing our derived equation, 3^(-x) = 3^(3x+6), with the given options, we can clearly see that option C matches our result. Options A and B have different exponent combinations, and option D is a simplified form of equating the exponents but not the equivalent exponential equation.

Option C, 3^(-x) = 3^(3x+6), is the equivalent equation.

Common Mistakes to Avoid

When working with exponential equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them.

Incorrectly applying the power of a power rule:

A common mistake is misapplying the power of a power rule. Remember that (a^m)n = a^(mn), not a^(m+n). For example, (3²⁾3 is 3^(23) = 3^6, not 3^(2+3) = 3^5. This distinction is crucial for simplifying exponential expressions correctly.

Forgetting the negative sign when dealing with reciprocals:

When expressing reciprocals as exponents, remember to include the negative sign. For instance, 1/3 is 3^(-1), not 3^1. Omitting the negative sign can lead to errors in the subsequent steps of the solution.

Failing to express both sides with the same base:

A key strategy in solving exponential equations is to express both sides with the same base. This allows you to equate the exponents and simplify the problem. Failing to do so can make the equation appear more complex than it actually is.

Arithmetic errors in simplifying exponents:

Simple arithmetic errors, such as incorrect multiplication or addition of exponents, can derail the entire solution. Double-check your calculations to ensure accuracy. This attention to detail is essential for arriving at the correct answer.

Misinterpreting the question:

Ensure you understand what the question is asking. In this case, we were asked to find an equivalent equation, not necessarily to solve for x. Misinterpreting the question can lead you down the wrong path.

By being mindful of these common mistakes, you can improve your accuracy and efficiency in solving exponential equations.

Real-World Applications of Exponential Equations

Exponential equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can provide a deeper appreciation for the importance of mastering exponential equations.

Finance:

Exponential functions are used to model compound interest, where the amount of money grows exponentially over time. Understanding exponential growth is crucial for making informed financial decisions, such as investments and loans.

Biology:

Exponential growth is observed in population dynamics, where the number of organisms can increase exponentially under ideal conditions. This is also relevant in understanding the spread of diseases, where the number of infected individuals can grow exponentially in the early stages of an outbreak.

Physics:

Exponential decay is used to model radioactive decay, where the amount of a radioactive substance decreases exponentially over time. This is essential in fields like nuclear physics and carbon dating.

Computer Science:

Exponential functions are used in algorithm analysis to describe the time complexity of certain algorithms. Understanding exponential time complexity is crucial for designing efficient algorithms.

Environmental Science:

Exponential growth and decay are used to model various environmental processes, such as the growth of pollution or the decay of pollutants in the environment.

These are just a few examples of the many real-world applications of exponential equations. The ability to work with exponential functions and equations is a valuable skill that can be applied in a wide range of fields.

Conclusion

In this article, we successfully identified the equation equivalent to (1/3)^x = 27^(x+2) by systematically applying the properties of exponents. We demonstrated the importance of expressing both sides of the equation with the same base and using the power of a power rule to simplify the equation. The equivalent equation, 3^(-x) = 3^(3x+6), was found to be option C. We also discussed common mistakes to avoid and highlighted the real-world applications of exponential equations. Mastering these concepts will not only improve your mathematical skills but also provide you with valuable tools for solving problems in various fields.

This step-by-step approach, combined with an understanding of common pitfalls and real-world applications, provides a comprehensive understanding of solving exponential equations. The ability to manipulate exponents and express numbers in different bases is a fundamental skill that opens doors to more advanced mathematical concepts. By carefully applying the rules of exponents, we can systematically reduce the complexity of the equation and arrive at its equivalent form. This exploration has not only provided the answer but also illuminated the underlying principles, making it easier to tackle similar problems in the future. Remember to practice these techniques and apply them to various problems to solidify your understanding.