Solving 9^x - 1 = 2 A Step-by-Step Guide
When dealing with mathematical problems, exponential equations often present a unique set of challenges and opportunities. These equations, characterized by a variable appearing in the exponent, require a specific approach to solve effectively. In this article, we will delve into the solution of the exponential equation 9^x - 1 = 2, providing a comprehensive step-by-step guide to understanding the process. Exponential equations are fundamental in various fields, including finance, physics, and computer science, making their mastery crucial for anyone pursuing studies or careers in these areas. This exploration will not only provide the answer to the equation but also illuminate the underlying principles and techniques applicable to a broader range of exponential problems. We will begin by isolating the exponential term, then use logarithms to solve for the variable x. This method is a cornerstone of exponential equation solving and will be explained in detail to ensure clarity. Whether you're a student tackling homework or a professional using these concepts in your work, this guide aims to enhance your understanding and problem-solving skills. By the end of this discussion, you should be confident in your ability to approach and solve similar exponential equations. Let's embark on this mathematical journey and uncover the solution together. Remember, practice is key to mastering these skills, so feel free to try more examples and explore different types of exponential equations.
Step 1: Isolate the Exponential Term
To effectively solve the equation 9^x - 1 = 2, the first crucial step involves isolating the exponential term. This means we need to manipulate the equation to get the term with the variable exponent, which is 9^x, by itself on one side of the equation. This isolation is essential because it allows us to apply further mathematical operations, such as logarithms, which are designed to deal with exponential expressions directly. The process begins by addressing the constant term on the same side as the exponential term. In our case, we have "-1" on the left side of the equation along with 9^x. To isolate 9^x, we need to eliminate this "-1". The mathematical operation that achieves this is adding 1 to both sides of the equation. This maintains the equality and moves us closer to our goal of isolating the exponential term. By adding 1 to both sides, we effectively cancel out the "-1" on the left side, leaving us with 9^x alone. On the right side, we perform the addition 2 + 1, which results in 3. Thus, after this step, our equation transforms from 9^x - 1 = 2 to 9^x = 3. This seemingly simple step is a cornerstone of solving exponential equations, as it sets the stage for the next phase of the solution. Now that we have isolated the exponential term, we can proceed with techniques that directly address the exponent, ultimately leading us to the value of x. In the following sections, we will explore how to apply logarithms to further simplify and solve the equation.
Step 2: Apply Logarithms
With the exponential term isolated as 9^x = 3, the next critical step is to apply logarithms. Logarithms are the inverse operation to exponentiation, making them an invaluable tool for solving equations where the variable is in the exponent. The fundamental principle here is that if a^b = c, then log_a(c) = b, where log_a represents the logarithm to the base a. Applying logarithms allows us to bring the exponent down, transforming the equation into a more manageable form. The choice of logarithm base is often a matter of convenience or the tools available. Common bases include base 10 (common logarithm), base e (natural logarithm, denoted as ln), and the base of the exponential term itself. In this case, we can choose any base, but using the natural logarithm (ln) is a versatile option that works well in most scenarios. Applying the natural logarithm to both sides of the equation 9^x = 3, we get ln(9^x) = ln(3). A key property of logarithms states that ln(a^b) = b * ln(a). Applying this property to the left side of our equation, we can rewrite ln(9^x) as x * ln(9). Thus, the equation becomes x * ln(9) = ln(3). This transformation is crucial because it brings the variable x out of the exponent, making it a coefficient. Now, the equation is a simple algebraic equation that we can solve for x. In the next step, we will focus on isolating x to find its value. Understanding the properties of logarithms is essential for effectively solving exponential equations, and this step demonstrates how they can be used to simplify and transform such equations into a solvable form.
Step 3: Solve for x
Having transformed the exponential equation into x * ln(9) = ln(3), our next objective is to isolate x and determine its value. This step involves basic algebraic manipulation. To get x by itself, we need to undo the multiplication by ln(9). The inverse operation of multiplication is division, so we will divide both sides of the equation by ln(9). This ensures that we maintain the equality while moving towards isolating x. Dividing both sides by ln(9), we get: (x * ln(9)) / ln(9) = ln(3) / ln(9). On the left side, ln(9) in the numerator and denominator cancels out, leaving us with x. Thus, we have x = ln(3) / ln(9). This expression provides the exact value of x, but it can be further simplified. We can rewrite 9 as 3^2, and using the property of logarithms, ln(a^b) = b * ln(a), we can express ln(9) as ln(3^2) = 2 * ln(3). Substituting this back into our equation, we get: x = ln(3) / (2 * ln(3)). Now, we see that ln(3) appears in both the numerator and the denominator, allowing us to cancel them out. This simplification leads us to: x = 1 / 2. Therefore, the solution to the equation 9^x - 1 = 2 is x = 1/2. This final step demonstrates how algebraic principles, combined with logarithmic properties, allow us to precisely solve for the variable in an exponential equation. The ability to manipulate equations and simplify expressions is a fundamental skill in mathematics, and this example illustrates its importance in the context of exponential equations.
Step 4: Verification
After obtaining a solution to any equation, it is crucial to verify its correctness. This step ensures that the value we found for x indeed satisfies the original equation. Verification not only confirms our solution but also helps in identifying any potential errors made during the solving process. In our case, we found x = 1/2 as the solution to the equation 9^x - 1 = 2. To verify this, we substitute x = 1/2 back into the original equation and check if both sides are equal. Substituting x = 1/2 into the equation, we get: 9^(1/2) - 1 = 2. Now, we need to evaluate 9^(1/2). Recall that a fractional exponent represents a root. Specifically, a^(1/2) is the square root of a. Therefore, 9^(1/2) is the square root of 9, which is 3. So, the equation becomes: 3 - 1 = 2. Performing the subtraction on the left side, we get: 2 = 2. Since the left side equals the right side, our solution x = 1/2 is indeed correct. This verification step underscores the importance of not just finding a solution but also ensuring its validity. It provides confidence in our answer and highlights the accuracy of the steps we took to solve the equation. In mathematical problem-solving, verification is a standard practice that should always be included, especially in examinations or when dealing with complex equations.
In conclusion, solving exponential equations like 9^x - 1 = 2 involves a systematic approach that combines algebraic manipulation with logarithmic properties. We began by isolating the exponential term, which is a critical first step in simplifying the equation. Then, we applied logarithms to both sides, using the property ln(a^b) = b * ln(a) to bring the variable out of the exponent. This transformation allowed us to convert the exponential equation into a linear equation that was easier to solve. By dividing both sides by ln(9), we isolated x and found its value to be ln(3) / ln(9), which simplified to 1/2. Finally, we verified our solution by substituting x = 1/2 back into the original equation, confirming its correctness. The process of solving exponential equations not only enhances our mathematical skills but also provides valuable insights into the relationship between exponential and logarithmic functions. Understanding these concepts is essential for various applications in science, engineering, and finance. Mastery of exponential equations requires practice and a solid understanding of the underlying principles. By following a step-by-step approach and verifying solutions, one can confidently tackle a wide range of exponential problems. This article has provided a comprehensive guide to solving one such equation, but the techniques and principles discussed can be applied to many others. Continued practice and exploration will further solidify your understanding and skills in this important area of mathematics.
