Identifying Points On The Graph Of F(x) = Log₉(x)
In this article, we will delve into the fascinating world of logarithmic functions, specifically focusing on the function f(x) = log₉(x). Our primary goal is to determine which points lie on the graph of this function. To achieve this, we will meticulously examine a set of given points and verify whether they satisfy the equation of the function. Understanding how to identify points on a logarithmic graph is crucial for grasping the behavior and properties of logarithmic functions, which are fundamental in various fields like mathematics, physics, and computer science. This exploration will not only enhance your analytical skills but also provide a solid foundation for more advanced mathematical concepts. We'll walk through each point step-by-step, providing clear explanations and calculations to ensure a comprehensive understanding of the process.
Before we begin verifying the points, it is essential to have a firm grasp of what a logarithmic function is. A logarithmic function is the inverse of an exponential function. In simpler terms, if we have an exponential equation b^y = x, the equivalent logarithmic equation is log_b(x) = y. Here, b is the base of the logarithm, x is the argument, and y is the exponent. The function we are dealing with, f(x) = log₉(x), has a base of 9. This means we are looking for the exponent to which we must raise 9 to get the value of x. Logarithmic functions are particularly useful for solving equations where the variable is in the exponent and for modeling phenomena that exhibit exponential growth or decay. Understanding the relationship between exponential and logarithmic forms is crucial for solving logarithmic problems and visualizing logarithmic graphs. The graph of a logarithmic function has several key characteristics, such as a vertical asymptote at x = 0 (for base > 1) and a slow rate of growth as x increases. By understanding these properties, we can better predict and interpret the behavior of the function.
Now, let's proceed to the task at hand: verifying which of the given points lie on the graph of f(x) = log₉(x). To do this, we will substitute the x and y coordinates of each point into the equation and check if the equation holds true. If the equation is satisfied, then the point lies on the graph; otherwise, it does not. This is a straightforward yet crucial process in coordinate geometry and function analysis. We will approach each point systematically, providing a clear step-by-step calculation and explanation. This method ensures accuracy and helps in understanding the relationship between a function's equation and its graphical representation. By the end of this section, you will have a clear understanding of how to verify points on any logarithmic graph.
Point 1: (-1/81, 2)
Let's start with the first point, (-1/81, 2). To determine if this point lies on the graph of f(x) = log₉(x), we substitute x = -1/81 and y = 2 into the equation. This gives us 2 = log₉(-1/81). Now, we need to check if this equation is true. Recall that the logarithm of a negative number is undefined in the realm of real numbers. In other words, there is no real number exponent to which we can raise 9 to get a negative result. Therefore, log₉(-1/81) is undefined, and the equation 2 = log₉(-1/81) is false. Consequently, the point (-1/81, 2) does not lie on the graph of f(x) = log₉(x). This demonstrates an important property of logarithmic functions: the argument of a logarithm must be positive. If the argument is zero or negative, the logarithm is undefined. This is because exponential functions always produce positive results, and logarithms are their inverses. Understanding this restriction is crucial when dealing with logarithmic functions and their graphs.
Point 2: (0, 1)
Next, we consider the point (0, 1). Substituting x = 0 and y = 1 into the equation f(x) = log₉(x), we get 1 = log₉(0). Similar to the previous case, we need to determine if this equation holds. The logarithm of zero is undefined for any base. This is because there is no exponent to which we can raise 9 (or any positive base other than 1) to obtain zero. The range of an exponential function is always positive, so its inverse, the logarithmic function, cannot have zero as its argument. Therefore, log₉(0) is undefined, and the equation 1 = log₉(0) is false. Thus, the point (0, 1) does not lie on the graph of f(x) = log₉(x). This reinforces the understanding that the argument of a logarithmic function must be strictly positive. The vertical asymptote of the logarithmic function at x = 0 visually represents this restriction on the domain.
Point 3: (1/9, -1)
Now, let's analyze the point (1/9, -1). Substituting x = 1/9 and y = -1 into the equation f(x) = log₉(x), we get -1 = log₉(1/9). To verify this, we need to determine if 9 raised to the power of -1 equals 1/9. Recall that a negative exponent indicates the reciprocal of the base raised to the positive exponent. In this case, 9⁻¹ = 1/9¹ = 1/9. Since 9⁻¹ indeed equals 1/9, the equation -1 = log₉(1/9) is true. Therefore, the point (1/9, -1) lies on the graph of f(x) = log₉(x). This example illustrates how to evaluate logarithms with fractional arguments and reinforces the connection between exponential and logarithmic forms. Understanding negative exponents is crucial for working with logarithmic functions and their inverses.
Point 4: (3, 243)
Moving on to the point (3, 243), we substitute x = 3 and y = 243 into the equation f(x) = log₉(x), resulting in 243 = log₉(3). To check this, we need to determine if 9 raised to the power of 243 equals 3. This is clearly not true, as 9 raised to a power will result in a much larger number than 3. To find the actual value of log₉(3), we need to determine the exponent to which we must raise 9 to get 3. Since 9^(1/2) = √9 = 3, we have log₉(3) = 1/2. Therefore, the equation 243 = log₉(3) is false, and the point (3, 243) does not lie on the graph of f(x) = log₉(x). This point serves as a reminder to carefully evaluate logarithmic expressions and to not assume a point lies on the graph without proper verification.
Point 5: (9, 1)
Now, let's consider the point (9, 1). Substituting x = 9 and y = 1 into the equation f(x) = log₉(x), we obtain 1 = log₉(9). We need to verify if this equation holds true. Recall that the logarithm of a number to the same base is always 1. In other words, log_b(b) = 1 for any valid base b. In this case, log₉(9) = 1 because 9 raised to the power of 1 equals 9. Therefore, the equation 1 = log₉(9) is true, and the point (9, 1) lies on the graph of f(x) = log₉(x). This illustrates a fundamental property of logarithms: the logarithm of the base to itself is always 1. This property is useful for quickly evaluating logarithmic expressions and identifying points on logarithmic graphs.
Point 6: (81, 2)
Finally, we examine the point (81, 2). Substituting x = 81 and y = 2 into the equation f(x) = log₉(x), we get 2 = log₉(81). To verify this, we need to determine if 9 raised to the power of 2 equals 81. Indeed, 9² = 9 * 9 = 81. Therefore, the equation 2 = log₉(81) is true, and the point (81, 2) lies on the graph of f(x) = log₉(x). This example demonstrates the direct application of the definition of a logarithm. By understanding the relationship between the base, exponent, and argument, we can easily verify whether a point lies on the graph of a logarithmic function. This skill is essential for solving logarithmic equations and analyzing logarithmic graphs.
In conclusion, we have meticulously examined six points to determine which of them lie on the graph of the logarithmic function f(x) = log₉(x). Through the process of substituting the x and y coordinates of each point into the equation and verifying the resulting statement, we have found that the points (1/9, -1), (9, 1), and (81, 2) lie on the graph. The points (-1/81, 2) and (0, 1) do not lie on the graph because the logarithm of a negative number and zero are undefined, respectively. The point (3, 243) does not lie on the graph as the logarithmic calculation did not match the given y-coordinate. This exercise underscores the importance of understanding the fundamental properties of logarithmic functions, including the domain restrictions and the relationship between exponential and logarithmic forms. By applying these principles, we can confidently determine whether a given point lies on the graph of a logarithmic function. This skill is invaluable in various mathematical contexts and real-world applications, such as solving exponential equations, modeling growth and decay phenomena, and analyzing data with logarithmic scales. The ability to accurately interpret and manipulate logarithmic functions is a cornerstone of mathematical literacy.