Solving 3^(x-1) + 3^(x-2) + 3^(x-3) = 13 An Exponential Equation

Introduction to Exponential Equations

In the realm of mathematics, exponential equations hold a significant place, particularly in fields like calculus, physics, and engineering. These equations, characterized by variables appearing in the exponents, often present intriguing challenges that demand a strong grasp of exponential properties and algebraic manipulation. This article delves into the step-by-step solution of a specific exponential equation: 3^(x-1) + 3^(x-2) + 3^(x-3) = 13. We'll break down each step, providing a clear understanding of the techniques involved and highlighting the underlying principles that govern exponential functions. Understanding how to solve exponential equations is crucial for various mathematical applications and real-world scenarios. From modeling population growth to calculating compound interest, exponential functions play a vital role. By mastering the techniques presented in this article, you'll gain a valuable tool for tackling complex mathematical problems and interpreting exponential phenomena.

Exponential equations may seem daunting at first glance, but with a systematic approach, they can be solved effectively. The key lies in leveraging the properties of exponents to simplify the equation and isolate the variable. This often involves techniques such as factoring, substitution, and the application of logarithmic functions. In this particular case, we'll explore a method that involves factoring out a common exponential term, which allows us to transform the equation into a more manageable form. This method is particularly useful when dealing with sums or differences of exponential terms with the same base. As we progress through the solution, we'll emphasize the importance of each step and provide explanations to ensure clarity. This will not only help you understand the solution to this specific equation but also equip you with the skills to solve similar exponential equations in the future. Remember, practice is essential in mathematics, so working through various examples will solidify your understanding and build your confidence. Let's embark on this journey of solving exponential equations and unlock the power of mathematical problem-solving.

Problem Statement: 3^(x-1) + 3^(x-2) + 3^(x-3) = 13

The given equation, 3^(x-1) + 3^(x-2) + 3^(x-3) = 13, is an exponential equation where the variable 'x' appears in the exponent. Our goal is to find the value(s) of 'x' that satisfy this equation. To achieve this, we will employ various algebraic techniques and properties of exponents to simplify the equation and ultimately isolate 'x'. This type of problem is common in algebra and calculus courses, highlighting the importance of mastering exponential equation-solving skills. Before diving into the solution, it's crucial to understand the fundamental properties of exponents. These properties will serve as our foundation for manipulating the equation and simplifying it into a more manageable form. For instance, the property a^(m+n) = a^m * a^n will be particularly useful in this case, as it allows us to break down the exponential terms into products of simpler terms. Similarly, the property a^(m-n) = a^m / a^n will enable us to express terms like 3^(x-1) as 3^x / 3^1. By applying these properties strategically, we can transform the equation into a form where factoring becomes possible. Factoring is a powerful technique for solving equations, as it allows us to express a sum or difference of terms as a product. This, in turn, can lead to simpler equations that are easier to solve. In the context of exponential equations, factoring often involves identifying a common exponential term that can be factored out. This is precisely the approach we will take in this solution.

The equation itself presents a sum of three exponential terms, each with a different exponent involving 'x'. This initially might seem complex, but by recognizing the common base (3) and applying the properties of exponents, we can rewrite the terms in a way that reveals a common factor. This common factor will be crucial for simplifying the equation. It's important to note that there are various approaches to solving exponential equations, but the method we'll use here is particularly effective for equations of this form. This approach involves identifying the smallest exponent among the terms and factoring out the corresponding exponential term. This strategy not only simplifies the equation but also avoids the introduction of fractions, which can sometimes complicate the process. Once we've factored out the common term, we'll be left with a simpler equation involving constants. This equation can then be solved using basic algebraic techniques. Finally, we'll need to check our solution(s) to ensure they satisfy the original equation. This is a crucial step in any equation-solving process, as it helps to identify any extraneous solutions that may have been introduced during the simplification process. With a clear understanding of the problem statement and the properties of exponents, we are now ready to embark on the solution.

Step-by-Step Solution

1. Identifying the Common Factor

Our initial equation is 3^(x-1) + 3^(x-2) + 3^(x-3) = 13. To simplify this, we'll first rewrite each term using the property a^(m-n) = a^m / a^n. This gives us:

(3^x / 3^1) + (3^x / 3^2) + (3^x / 3^3) = 13

Now, observe that 3^x is a common factor in all three terms. This is a crucial observation that allows us to simplify the equation significantly. Identifying common factors is a fundamental skill in algebra and is particularly useful in solving equations. In this case, recognizing the common factor of 3^x allows us to transform the equation from a sum of fractions to a product, which is much easier to handle. Factoring out the common factor is a powerful technique that can be applied to various types of equations, not just exponential equations. It's a key step in simplifying expressions and solving for unknown variables. The ability to identify common factors quickly and efficiently is a valuable asset in any mathematical problem-solving scenario.

Before we proceed with factoring, let's clarify why this step is so important. By factoring out the common term, we are essentially reversing the distributive property. This allows us to group the terms that contain 'x' together, making it easier to isolate the variable. In the context of exponential equations, factoring often involves identifying a common exponential term, as we've done here. This common term can then be factored out, leaving us with a simpler equation that we can solve using algebraic techniques. The identification of the common factor is not always straightforward, and it may require some manipulation of the terms. However, with practice, you'll become more adept at recognizing patterns and identifying common factors quickly.

2. Factoring out 3^x

Next, we factor out 3^x from the left side of the equation:

3^x * (1/3 + 1/9 + 1/27) = 13

This step is the heart of our simplification process. By factoring out 3^x, we've transformed the equation into a product of 3^x and a constant expression. This makes it much easier to isolate 3^x and subsequently solve for 'x'. The expression inside the parentheses, (1/3 + 1/9 + 1/27), is a simple arithmetic series that can be easily evaluated. This simplification is a direct consequence of factoring out the common term. Factoring is a powerful technique in algebra, allowing us to rewrite expressions in a more manageable form. It's a skill that is essential for solving equations, simplifying expressions, and performing various other mathematical operations. In this case, factoring has allowed us to transform a complex exponential equation into a simpler one that can be solved using basic algebraic techniques. The key to successful factoring is to identify the common factors and apply the distributive property in reverse. This requires a good understanding of algebraic manipulation and the properties of exponents.

Now, let's delve a bit deeper into the mechanics of factoring. When we factor out 3^x, we are essentially dividing each term in the original equation by 3^x and placing it outside the parentheses. This is the reverse of the distributive property, which states that a(b + c) = ab + ac. In our case, we are going from ab + ac + ad = a(b + c + d). This process is fundamental to simplifying algebraic expressions and solving equations. The ability to factor effectively is a crucial skill in mathematics, and it is used extensively in various branches of the subject, including algebra, calculus, and differential equations. By mastering factoring techniques, you will be well-equipped to tackle a wide range of mathematical problems.

3. Simplifying the Constant Term

Now, we simplify the constant term inside the parentheses:

1/3 + 1/9 + 1/27 = 9/27 + 3/27 + 1/27 = 13/27

So, our equation becomes:

3^x * (13/27) = 13

This step involves basic arithmetic operations of adding fractions. Finding a common denominator is essential for adding fractions, and in this case, the least common denominator is 27. Once we've found the common denominator, we can simply add the numerators. This simplification is a crucial step in isolating 3^x. By combining the fractions into a single term, we've made the equation more concise and easier to manipulate. Arithmetic operations are the foundation of mathematics, and a strong understanding of these operations is essential for solving any mathematical problem. In this case, the ability to add fractions efficiently is crucial for simplifying the equation and progressing towards the solution.

Let's take a moment to appreciate the progress we've made so far. We started with a complex exponential equation, and through the steps of identifying a common factor, factoring it out, and simplifying the constant term, we've transformed the equation into a much simpler form. This highlights the power of algebraic manipulation in solving mathematical problems. Each step we've taken has brought us closer to isolating the variable 'x'. This systematic approach is key to solving any equation, whether it's an exponential equation, a linear equation, or a quadratic equation. By breaking down the problem into smaller, manageable steps, we can tackle even the most complex equations with confidence.

4. Isolating 3^x

To isolate 3^x, we multiply both sides of the equation by the reciprocal of 13/27, which is 27/13:

3^x = 13 * (27/13)

3^x = 27

This step is a standard algebraic manipulation technique used to isolate a variable. Multiplying both sides of an equation by the same non-zero value preserves the equality. In this case, multiplying by the reciprocal of 13/27 cancels out the 13/27 on the left side, leaving us with 3^x isolated. This is a crucial step in solving for 'x'. Isolating the variable is a fundamental goal in solving any equation. It allows us to express the variable in terms of known constants, thus providing the solution to the equation. The technique of multiplying by the reciprocal is particularly useful when dealing with fractions, as it effectively eliminates the fraction from the equation. This simplifies the equation and makes it easier to solve.

Now that we have isolated 3^x, we are one step closer to finding the value of 'x'. The equation 3^x = 27 is a much simpler exponential equation than the original equation. This simplification is a testament to the effectiveness of the algebraic techniques we've employed so far. We've successfully transformed a complex equation into a manageable one. This process of simplification is a common theme in mathematics. By breaking down complex problems into smaller, simpler steps, we can make them more approachable and ultimately solve them. The ability to simplify equations and expressions is a valuable skill in mathematics, and it is essential for success in higher-level mathematics courses.

5. Solving for x

We recognize that 27 can be written as 3^3:

3^x = 3^3

Since the bases are the same, we can equate the exponents:

x = 3

This is the final step in solving for 'x'. By recognizing that 27 can be expressed as 3 raised to the power of 3, we can directly equate the exponents. This is a fundamental property of exponential functions: if a^m = a^n, then m = n, provided that a is not equal to 0, 1, or -1. Applying this property allows us to solve for 'x' directly. The ability to recognize powers of numbers is a valuable skill in mathematics. It allows us to simplify expressions and solve equations more efficiently. In this case, recognizing that 27 is 3 cubed is crucial for finding the solution to the equation. This recognition comes with practice and familiarity with common powers of numbers.

The solution x = 3 is the value that satisfies the original equation. This means that when we substitute 3 for 'x' in the original equation, the equation holds true. Let's verify this solution in the next step to ensure its correctness. Solving for 'x' is the ultimate goal in this problem, and we have successfully achieved it. The process of solving for 'x' has involved a series of algebraic manipulations and the application of exponential properties. Each step has been carefully chosen to simplify the equation and bring us closer to the solution. This systematic approach is key to solving any mathematical problem, and it is a valuable skill to develop.

6. Verification

To verify our solution, we substitute x = 3 back into the original equation:

3^(3-1) + 3^(3-2) + 3^(3-3) = 3^2 + 3^1 + 3^0 = 9 + 3 + 1 = 13

Since the equation holds true, our solution is correct.

This final step is crucial in any equation-solving process. Verification ensures that the solution we've obtained is indeed correct and that no errors were made during the simplification process. Substituting the solution back into the original equation allows us to check if the equation holds true. If the equation does not hold true, then we know that there is an error in our solution, and we need to re-examine our steps. Verification is a safeguard against mistakes and a way to build confidence in our solution. In this case, substituting x = 3 back into the original equation, we found that the equation holds true, confirming that our solution is correct. This final step provides closure to the problem-solving process and reinforces the correctness of our answer. The habit of verifying solutions is a valuable one to develop, as it ensures accuracy and builds confidence in one's mathematical abilities.

Conclusion

We have successfully solved the exponential equation 3^(x-1) + 3^(x-2) + 3^(x-3) = 13 and found the solution to be x = 3. This process involved several key steps, including identifying a common factor, factoring it out, simplifying the constant term, isolating 3^x, and finally solving for 'x'. Each step was carefully explained, highlighting the underlying algebraic principles and properties of exponents. Furthermore, we verified our solution to ensure its correctness. This thorough approach demonstrates the importance of a systematic and methodical approach to solving mathematical problems.

Solving exponential equations is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, and finance. The techniques used in this article can be applied to a wide range of exponential equations, making it a valuable tool for any student or professional working with mathematical models. By mastering these techniques, you'll be well-equipped to tackle complex mathematical problems and interpret exponential phenomena. The key takeaways from this solution are the importance of identifying common factors, applying the properties of exponents, and systematically simplifying the equation. These skills are transferable to other areas of mathematics and problem-solving in general. Remember, practice is essential for mastering any mathematical skill. By working through various examples and applying the techniques learned in this article, you'll solidify your understanding and build your confidence in solving exponential equations. This journey of solving exponential equations is not just about finding the answer; it's about developing a deeper understanding of mathematical principles and honing problem-solving skills that will serve you well in various aspects of life.