Exploring The Function Y=log₃(x-2) A Comprehensive Guide
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In the realm of mathematics, understanding functions is paramount. Among the various types of functions, logarithmic functions hold a special place due to their unique properties and applications. This comprehensive guide delves into the intricacies of the function y = log₃(x - 2), aiming to provide a thorough understanding of its behavior and characteristics. We will embark on a journey to complete a table of values for this function, exploring its domain, range, and graphical representation. By the end of this exploration, you will have a solid grasp of this logarithmic function and its significance in the broader mathematical landscape.
Understanding Logarithmic Functions
Before we dive into the specifics of y = log₃(x - 2), let's first lay the groundwork by understanding the fundamental concepts of logarithmic functions. A logarithmic function is the inverse of an exponential function. In simpler terms, if we have an exponential equation like bˣ = y, its logarithmic form is logb(y) = x. Here, b is the base of the logarithm, y is the argument, and x is the exponent. The logarithm essentially answers the question: "To what power must we raise the base b to obtain the value y?"
Logarithmic functions are characterized by several key properties. First, the domain of a logarithmic function logb(x) is restricted to positive values of x (i.e., x > 0). This is because we cannot raise a positive base to any power and obtain a non-positive result. Second, the range of a logarithmic function is all real numbers. This means that the logarithm can take on any real value, positive, negative, or zero. Third, the graph of a logarithmic function has a vertical asymptote at x = 0. This is because the function approaches infinity (or negative infinity) as x approaches zero. Finally, logarithmic functions are either increasing or decreasing, depending on the base b. If b > 1, the function is increasing, and if 0 < b < 1, the function is decreasing.
The Function y = log₃(x - 2): A Detailed Analysis
Now that we have a firm understanding of logarithmic functions in general, let's turn our attention to the specific function y = log₃(x - 2). This function is a logarithmic function with base 3, and its argument is (x - 2). The presence of (x - 2) inside the logarithm introduces a horizontal shift to the graph of the basic logarithmic function y = log₃(x). This shift is crucial in determining the domain and behavior of the function.
To determine the domain of y = log₃(x - 2), we need to ensure that the argument (x - 2) is positive. This means that x - 2 > 0, which implies x > 2. Therefore, the domain of the function is all real numbers greater than 2. In interval notation, the domain is (2, ∞). This restriction on the domain has a significant impact on the graph of the function, as it introduces a vertical asymptote at x = 2. The graph will approach this line but never actually touch it.
The range of y = log₃(x - 2), like all logarithmic functions, is all real numbers. This means that y can take on any value, positive, negative, or zero. As x approaches 2 from the right, y approaches negative infinity. As x increases, y also increases, albeit at a decreasing rate. This behavior is characteristic of logarithmic functions with a base greater than 1.
Completing the Table of Values
Our primary goal is to complete the table of values for the function y = log₃(x - 2). This involves finding the corresponding y-values for given x-values. Let's consider the following table:
| x | y |
|---|---|
| 19/9 | |
| 7 | |
| 3 | |
| 5 |
To complete the table, we will substitute each x-value into the function y = log₃(x - 2) and solve for y. This process will demonstrate how the function behaves for different inputs and provide a clearer picture of its graph.
Calculating y for x = 19/9
First, let's find the value of y when x = 19/9. Substituting this value into the function, we get:
y = log₃((19/9) - 2)
To simplify the expression inside the logarithm, we need to find a common denominator for 19/9 and 2. Since 2 can be written as 18/9, we have:
y = log₃((19/9) - (18/9))
y = log₃(1/9)
Now, we need to determine to what power we must raise 3 to obtain 1/9. Since 1/9 is equal to 3-2, we have:
y = log₃(3-2)
Using the property of logarithms that logb(bx) = x, we get:
y = -2
Therefore, when x = 19/9, y = -2.
Calculating y for x = 7
Next, let's find the value of y when x = 7. Substituting this value into the function, we get:
y = log₃(7 - 2)
y = log₃(5)
The value of log₃(5) is not a simple integer. We can use a calculator to approximate its value. Using a calculator, we find that:
log₃(5) ≈ 1.465
Therefore, when x = 7, y ≈ 1.465.
Calculating y for x = 3
Now, let's find the value of y when x = 3. Substituting this value into the function, we get:
y = log₃(3 - 2)
y = log₃(1)
We know that any number raised to the power of 0 is equal to 1. Therefore, log₃(1) = 0.
Thus, when x = 3, y = 0.
Calculating y for x = 5
Finally, let's find the value of y when x = 5. Substituting this value into the function, we get:
y = log₃(5 - 2)
y = log₃(3)
We know that 3 raised to the power of 1 is equal to 3. Therefore, log₃(3) = 1.
Thus, when x = 5, y = 1.
The Completed Table
Now that we have calculated the y-values for each given x-value, we can complete the table:
| x | y |
|---|---|
| 19/9 | -2 |
| 7 | ≈ 1.465 |
| 3 | 0 |
| 5 | 1 |
This completed table provides a set of points that lie on the graph of the function y = log₃(x - 2). These points can be used to sketch the graph of the function and visualize its behavior.
Graphing the Function y = log₃(x - 2)
To further enhance our understanding of the function y = log₃(x - 2), let's explore its graphical representation. We can plot the points from the completed table on a coordinate plane and connect them to form the graph of the function. Additionally, we need to consider the domain and the vertical asymptote to accurately sketch the graph.
We know that the domain of the function is x > 2, and there is a vertical asymptote at x = 2. This means that the graph will approach the line x = 2 but never cross it. The points from the completed table are (19/9, -2), (7, 1.465), (3, 0), and (5, 1). Plotting these points and considering the asymptote, we can sketch the graph of the function. The graph will start near the asymptote at x = 2 and gradually increase as x increases. It will pass through the points we plotted and continue to increase without bound.
The graph of y = log₃(x - 2) is a transformation of the basic logarithmic function y = log₃(x). The transformation involves a horizontal shift of 2 units to the right. This shift is due to the (x - 2) term inside the logarithm. Understanding these transformations is crucial for quickly sketching the graphs of logarithmic functions.
Applications of Logarithmic Functions
Logarithmic functions are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including:
- Science: Logarithms are used in chemistry to measure pH levels, in physics to measure sound intensity (decibels), and in geology to measure the magnitude of earthquakes (the Richter scale).
- Finance: Logarithms are used to calculate compound interest and to model financial growth.
- Computer science: Logarithms are used in algorithms to analyze the efficiency of search and sorting algorithms.
- Information theory: Logarithms are used to measure information entropy.
The function y = log₃(x - 2), while a specific example, embodies the general properties and applications of logarithmic functions. Understanding this function provides a solid foundation for exploring more complex logarithmic models and their applications in various fields.
Conclusion
In this comprehensive guide, we have delved into the function y = log₃(x - 2). We started by understanding the fundamental concepts of logarithmic functions, including their properties, domain, and range. We then focused on the specific function, determining its domain, identifying its vertical asymptote, and completing a table of values. By plotting these values, we were able to sketch the graph of the function and visualize its behavior. Finally, we explored the various real-world applications of logarithmic functions, highlighting their importance in science, finance, computer science, and information theory.
By working through this guide, you have gained a deeper understanding of the function y = log₃(x - 2) and logarithmic functions in general. You are now equipped with the knowledge and skills to analyze and graph other logarithmic functions and appreciate their significance in the broader mathematical landscape.
