Graphing The Logarithmic Function G(x) = Log₂(x-1) Domain And Range

Jul 11, 2025 by ADMIN 68 views

Graphing the Logarithmic Function g(x) = log₂(x-1): A Comprehensive Guide

Introduction

In this comprehensive guide, we will delve into the process of graphing the logarithmic function g(x) = log₂(x-1). This involves plotting key points, identifying the asymptote, and utilizing graphing tools. We will also determine the domain and range of the function, providing a complete understanding of its behavior. Understanding logarithmic functions is crucial in various fields, including mathematics, physics, and computer science. Logarithmic functions are the inverse of exponential functions, and they help in solving equations where the variable is in the exponent. In our specific case, g(x) = log₂(x-1) represents a logarithmic function with a base of 2 and a horizontal shift. To accurately graph this function, we need to understand the transformations applied to the basic logarithmic function log₂(x). The (x-1) term inside the logarithm indicates a horizontal shift of the graph to the right by 1 unit. This shift affects the vertical asymptote of the function, which is a critical feature in graphing logarithmic functions. The domain and range of the function are also crucial aspects to consider. The domain represents the set of all possible input values (x-values) for which the function is defined, while the range represents the set of all possible output values (y-values). For logarithmic functions, the domain is restricted by the argument of the logarithm, which must be greater than zero. The range, however, is all real numbers. By carefully plotting points, identifying the asymptote, and understanding the domain and range, we can accurately graph the logarithmic function g(x) = log₂(x-1) and gain a deeper insight into its behavior.

1. Plotting Points on the Graph

To begin graphing g(x) = log₂(x-1), we need to plot at least two points. Choosing appropriate x-values is crucial for simplifying the logarithmic expression. Key points can be found by selecting x-values that make the argument of the logarithm a power of the base. In this case, the base is 2, so we want (x-1) to be equal to powers of 2, such as 1, 2, 4, etc. First, let's find the vertical asymptote, which is a vertical line that the graph approaches but never touches. For g(x) = log₂(x-1), the vertical asymptote occurs when the argument of the logarithm (x-1) is equal to zero. Solving x-1 = 0, we find that x = 1. Thus, the vertical asymptote is the line x = 1. This line will guide our plotting and ensure that we accurately represent the function's behavior near the asymptote. Next, we choose x-values to plot. If we set x-1 = 1, then x = 2. Substituting x = 2 into the function, we get g(2) = log₂(2-1) = log₂(1) = 0. This gives us the point (2, 0), which is an important point on the graph. It is also the x-intercept of the function, where the graph crosses the x-axis. Next, we choose another x-value. If we set x-1 = 2, then x = 3. Substituting x = 3 into the function, we get g(3) = log₂(3-1) = log₂(2) = 1. This gives us the point (3, 1), another point on the graph. By plotting these points, we start to see the shape of the logarithmic function. To further refine our graph, we can choose additional points. If we set x-1 = 4, then x = 5. Substituting x = 5 into the function, we get g(5) = log₂(5-1) = log₂(4) = 2. This gives us the point (5, 2). Similarly, if we set x-1 = 1/2, then x = 3/2. Substituting x = 3/2 into the function, we get g(3/2) = log₂(3/2-1) = log₂(1/2) = -1. This gives us the point (3/2, -1). By plotting these additional points, we gain a more detailed understanding of the function's curve and its behavior as it approaches the vertical asymptote.

2. Drawing the Asymptote

The asymptote is a crucial feature of logarithmic functions. It is a line that the graph approaches but never intersects. For the function g(x) = log₂(x-1), the argument of the logarithm, (x-1), must be greater than zero. This is because the logarithm of a non-positive number is undefined. Therefore, we have the inequality x-1 > 0, which simplifies to x > 1. This inequality tells us that the function is defined for all x-values greater than 1. The vertical asymptote occurs at the boundary of this domain restriction. To find the equation of the vertical asymptote, we set the argument of the logarithm equal to zero and solve for x: x-1 = 0, which gives us x = 1. Thus, the vertical asymptote is the line x = 1. This means that the graph of the function g(x) = log₂(x-1) will approach the line x = 1 but will never cross it. The vertical asymptote serves as a guide when sketching the graph of the function. As x approaches 1 from the right, the function values approach negative infinity. This behavior is characteristic of logarithmic functions near their vertical asymptotes. To draw the asymptote, we sketch a dashed vertical line at x = 1 on the coordinate plane. This dashed line indicates that the function gets arbitrarily close to this line but never touches it. The asymptote is an essential reference point for accurately graphing the function. It helps to define the boundaries of the graph and provides crucial information about the function's behavior. When plotting points, we should ensure that the graph approaches the asymptote as x approaches 1 from the right. The vertical asymptote also helps in understanding the domain of the function, as it visually represents the lower bound of the possible x-values. In summary, drawing the asymptote x = 1 is a critical step in graphing g(x) = log₂(x-1). It provides a clear boundary for the graph and helps to accurately represent the function's behavior near this boundary.

3. Using Graphing Tools

After plotting points and drawing the asymptote, utilizing graphing tools can help to visualize the logarithmic function g(x) = log₂(x-1) more accurately. Graphing calculators and online graphing utilities, such as Desmos or GeoGebra, provide a convenient way to plot functions and explore their properties. These tools allow you to input the function g(x) = log₂(x-1) and generate its graph automatically. When using a graphing calculator, you typically enter the function into the function editor and then adjust the viewing window to display the relevant portion of the graph. It's important to set the window appropriately to see the key features of the function, such as the vertical asymptote and the points we plotted earlier. For g(x) = log₂(x-1), you might set the x-axis range to be from 0 to 10 and the y-axis range to be from -5 to 5. This will give you a clear view of the function's behavior near the asymptote and its overall shape. Online graphing utilities like Desmos and GeoGebra offer similar functionality but with the added benefit of being accessible from any device with an internet connection. These tools often have user-friendly interfaces that make it easy to input functions, adjust the graph's appearance, and analyze its properties. In addition to plotting the function, graphing tools can also help you identify key features such as the x-intercept, y-intercept (if any), and the asymptote. They can also provide a more precise view of the function's behavior as it approaches the asymptote. To use the graphing tool effectively, input the function g(x) = log₂(x-1). Ensure that you correctly enter the base of the logarithm, which is 2 in this case. Some tools may require you to use the change of base formula to express the logarithm in terms of the natural logarithm (ln) or the common logarithm (log). Once the function is plotted, you can zoom in and out to examine the graph in more detail. You can also trace along the graph to find specific points and their coordinates. Graphing tools are invaluable for verifying the accuracy of your hand-drawn graph and for gaining a deeper understanding of the function's behavior. They allow you to see the overall shape of the graph, identify key features, and explore the function's properties in a dynamic and interactive way. By using these tools, you can confirm that your plotted points and asymptote are consistent with the function's graph, ensuring that your understanding of the function is accurate and complete.

4. Determining the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the logarithmic function g(x) = log₂(x-1), the domain is restricted by the argument of the logarithm, which must be greater than zero. This is because the logarithm of a non-positive number is undefined. To find the domain of g(x) = log₂(x-1), we set the argument (x-1) greater than zero: x-1 > 0. Solving this inequality for x, we add 1 to both sides: x > 1. This inequality tells us that the function is defined for all x-values greater than 1. Therefore, the domain of g(x) = log₂(x-1) is the interval (1, ∞). This interval notation indicates that the domain includes all real numbers greater than 1, but it does not include 1 itself. The parenthesis around 1 signifies that 1 is not included in the domain, which is consistent with the fact that the logarithm of zero is undefined. The symbol ∞ represents infinity, indicating that the domain extends indefinitely to the right. Understanding the domain is crucial for accurately graphing and interpreting the function. It tells us where the function is defined and where it is not. In the case of g(x) = log₂(x-1), the domain restriction x > 1 corresponds to the vertical asymptote at x = 1. The graph of the function will approach the vertical asymptote but will never cross it, reflecting the fact that the function is not defined for x ≤ 1. The domain also helps us to understand the behavior of the function as x approaches the boundary of the domain. As x approaches 1 from the right, the function values approach negative infinity. This behavior is characteristic of logarithmic functions near their vertical asymptotes. In summary, the domain of g(x) = log₂(x-1) is (1, ∞), which means that the function is defined for all x-values greater than 1. This domain restriction is a key feature of logarithmic functions and is essential for understanding their graphs and properties. Knowing the domain allows us to accurately plot the function and interpret its behavior within the defined range of input values.

5. Determining the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For logarithmic functions, the range is typically all real numbers. To determine the range of g(x) = log₂(x-1), we consider the behavior of the function as x varies within its domain. The domain of g(x) = log₂(x-1) is (1, ∞), as we established earlier. This means that x can take any value greater than 1. As x approaches 1 from the right, the argument of the logarithm (x-1) approaches 0, and the function values approach negative infinity. This is because the logarithm of a very small positive number is a large negative number. As x increases without bound, the argument (x-1) also increases without bound, and the function values increase without bound as well. This is because the logarithm of a large number is a positive number. Since the function values can take on any real number, the range of g(x) = log₂(x-1) is all real numbers. This can be expressed in interval notation as (-∞, ∞). The parentheses around (-∞) and (∞) indicate that these values are not included in the range, as the function never actually reaches infinity or negative infinity. Understanding the range is important for fully characterizing the behavior of a function. It tells us the possible output values that the function can produce, which can be useful in various applications. In the case of g(x) = log₂(x-1), the range of all real numbers means that the function can take on any y-value. This is a characteristic property of logarithmic functions, which have no upper or lower bound on their output values. The range also helps us to visualize the graph of the function. Since the range is all real numbers, the graph of g(x) = log₂(x-1) extends vertically from negative infinity to positive infinity. This is consistent with the shape of logarithmic functions, which have a vertical asymptote and then gradually increase or decrease as x moves away from the asymptote. In summary, the range of g(x) = log₂(x-1) is (-∞, ∞), which means that the function can produce any real number as an output value. This range is a key feature of logarithmic functions and is essential for understanding their graphs and properties. Knowing the range allows us to fully characterize the function's behavior and interpret its output values within the context of the input values.

Conclusion

Graphing the logarithmic function g(x) = log₂(x-1) involves several key steps: plotting points, drawing the asymptote, using graphing tools, and determining the domain and range. By carefully following these steps, we can accurately visualize and understand the behavior of the function. Plotting points helps us to establish the basic shape of the graph and identify key features such as the x-intercept. Drawing the vertical asymptote at x = 1 is crucial for understanding the function's behavior near the boundary of its domain. Using graphing tools such as graphing calculators and online utilities allows us to verify our hand-drawn graph and explore the function's properties in more detail. Determining the domain and range provides a complete understanding of the function's input and output values. The domain of g(x) = log₂(x-1) is (1, ∞), indicating that the function is defined for all x-values greater than 1. The range is (-∞, ∞), meaning that the function can produce any real number as an output value. Understanding logarithmic functions is essential in various fields, including mathematics, physics, and computer science. These functions are the inverse of exponential functions and are used to solve equations where the variable is in the exponent. By mastering the techniques for graphing logarithmic functions, we can gain a deeper appreciation for their properties and applications. In summary, graphing g(x) = log₂(x-1) is a comprehensive process that requires a combination of analytical and graphical skills. By plotting points, drawing the asymptote, using graphing tools, and determining the domain and range, we can fully understand and visualize the behavior of this logarithmic function. This understanding is crucial for further studies in mathematics and related fields.