Finding The Graph Representing The Solution To 4^(x-3) = 8

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Introduction

In the realm of mathematics, deciphering equations and visualizing their solutions graphically is a fundamental skill. This article delves into the process of identifying the graph that represents the solution to the exponential equation 4^(x-3) = 8. We will embark on a step-by-step journey, exploring the underlying concepts, employing algebraic manipulation, and ultimately connecting the solution to its graphical representation. Our focus will be on providing a clear and comprehensive understanding of the equation and its solution, ensuring that readers can confidently tackle similar problems in the future. We will break down the equation, discuss the properties of exponential functions, and demonstrate how to solve for the unknown variable, x. The goal is to empower readers with the knowledge and skills to not only solve this specific equation but also to understand the broader principles of solving exponential equations and interpreting their graphical solutions. By the end of this article, you will have a firm grasp of the relationship between equations, solutions, and their graphical representations.

Understanding the Equation: 4^(x-3) = 8

At the heart of our exploration lies the exponential equation 4^(x-3) = 8. To effectively solve this equation and pinpoint its graphical solution, we must first dissect its components and understand the roles they play. Exponential equations involve a constant base raised to a variable exponent, and in this case, the base is 4 and the exponent is (x-3). The equation states that 4 raised to the power of (x-3) is equal to 8. This seemingly simple statement holds a wealth of mathematical information, and our task is to unravel it. The key to solving exponential equations often lies in manipulating the equation to have the same base on both sides. This allows us to equate the exponents and solve for the unknown variable. Understanding the properties of exponents is crucial in this process. For instance, we can express both 4 and 8 as powers of the same base, which will be a pivotal step in simplifying the equation. This initial understanding sets the stage for our algebraic journey, paving the way for us to isolate the variable and determine the solution. The ability to recognize and manipulate exponential expressions is a cornerstone of mathematical proficiency, and this section aims to solidify that foundation.

Solving the Equation Algebraically

To pinpoint the solution to 4^(x-3) = 8, we'll employ the power of algebraic manipulation. Our primary goal is to isolate the variable 'x'. The first step involves expressing both sides of the equation with the same base. Recognizing that both 4 and 8 are powers of 2, we can rewrite the equation as (22)(x-3) = 2^3. This transformation is crucial as it allows us to directly compare the exponents. Next, we apply the power of a power rule, which states that (am)n = a^(mn). Applying this rule to the left side of the equation, we get 2^(2(x-3)) = 2^3. Now, we have the same base on both sides, enabling us to equate the exponents: 2*(x-3) = 3. This simplifies the equation significantly, transforming it into a linear equation that is much easier to solve. Expanding the left side, we get 2x - 6 = 3. Adding 6 to both sides, we have 2x = 9. Finally, dividing both sides by 2, we arrive at the solution: x = 9/2 or 4.5. This algebraic solution provides us with a precise value for x that satisfies the original equation. This value will be the key to identifying the correct graph representing the solution.

Graphical Representation of the Solution

Having determined the solution x = 4.5 algebraically, we now turn our attention to its graphical representation. The equation 4^(x-3) = 8 can be interpreted graphically as the intersection point of two functions: y = 4^(x-3) and y = 8. The first function, y = 4^(x-3), is an exponential function, which exhibits a characteristic curve that either increases or decreases rapidly. In this case, since the base (4) is greater than 1, the function is increasing. The second function, y = 8, is a horizontal line, representing a constant value of y. The point where these two graphs intersect corresponds to the x-value that satisfies the equation 4^(x-3) = 8. Therefore, the x-coordinate of the intersection point will be our solution, x = 4.5. To visualize this, imagine plotting both functions on a coordinate plane. The exponential curve of y = 4^(x-3) will start relatively flat and then rise sharply, while the horizontal line y = 8 will cut across the graph. The point of intersection will be where the y-value of the exponential function equals 8, and the corresponding x-value will be 4.5. This graphical interpretation provides a visual confirmation of our algebraic solution and enhances our understanding of the relationship between equations and their graphical representations. Identifying the correct graph involves locating the intersection point and verifying that its x-coordinate matches our calculated solution.

Identifying the Correct Graph

The culmination of our algebraic and conceptual understanding leads us to the crucial task of identifying the correct graph that visually represents the solution to 4^(x-3) = 8. We know that the solution, x = 4.5, corresponds to the point where the graphs of y = 4^(x-3) and y = 8 intersect. Therefore, the correct graph will be the one that depicts these two functions and clearly shows their intersection at x = 4.5. To effectively identify the correct graph, we need to look for several key features. First, we need to confirm that the graph includes an exponential curve that resembles the shape of y = 4^(x-3). This curve should be increasing, as the base (4) is greater than 1. Additionally, the graph should include a horizontal line representing y = 8. The most critical feature, however, is the intersection point. The x-coordinate of this point must be 4.5. We can visually estimate this by checking the scale of the x-axis and ensuring that the intersection occurs midway between 4 and 5. By carefully examining the provided graphs and comparing them to these criteria, we can confidently pinpoint the one that accurately represents the solution to the equation. This process reinforces the connection between algebraic solutions and their visual representations, solidifying our understanding of mathematical concepts.

Conclusion

In this comprehensive exploration, we have successfully navigated the process of identifying the graph that represents the solution to the equation 4^(x-3) = 8. We began by dissecting the equation and understanding its exponential nature. Through algebraic manipulation, we methodically solved for x, arriving at the solution x = 4.5. We then transitioned to the graphical interpretation, recognizing that the solution corresponds to the intersection point of the functions y = 4^(x-3) and y = 8. By carefully analyzing the key features of the graphs, we learned how to identify the one that accurately depicts this intersection at x = 4.5. This journey has not only provided a solution to the specific problem but has also reinforced fundamental mathematical principles, including the properties of exponents, algebraic problem-solving techniques, and the connection between equations and their graphical representations. The ability to seamlessly transition between algebraic and graphical perspectives is a hallmark of mathematical proficiency, and this article has aimed to foster that skill in our readers. By understanding the underlying concepts and applying a systematic approach, you can confidently tackle similar problems and deepen your understanding of the mathematical world.