Inverse of Exponential Functions are Logarithmic Functions, Lecture notes of Elementary Mathematics

Download Inverse of Exponential Functions are Logarithmic Functions and more Lecture notes Elementary Mathematics in PDF only on Docsity! Math Instructional Framework Full Name Math III Unit 3 Lesson 2 Time Frame Unit Name Logarithmic Functions as Inverses of Exponential Functions Learning Task/Topics/ Themes Task 2: How long Does It Take? Task 3: The Population of Exponentia Task 4: Modeling Natural Phenomena on Earth Culminating Task: Traveling to Exponentia Standards and Elements MM3A2. Students will explore logarithmic functions as inverses of exponential functions. c. Define logarithmic functions as inverses of exponential functions. Lesson Essential Questions How can you graph the inverse of an exponential function? Activator PROBLEM 2.Task 3: The Population of Exponentia (Problem 1 could be completed prior) Work Session Inverse of Exponential Functions are Logarithmic Functions A Graph the inverse of exponential functions. B Graph logarithmic functions. See Notes Below. VOCABULARY Asymptote: A line or curve that describes the end behavior of the graph. A graph never crosses a vertical asymptote but it may cross a horizontal or oblique asymptote. Common logarithm: A logarithm with a base of 10. A common logarithm is the power, a, such that 10 a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 10 2 = 100. Exponential functions: A function of the form y = a·bx where a > 0 and either 0 < b < 1 or b > 1. Logarithmic functions: A function of the form y = logbx, with b  1 and b and x both positive. A logarithmic function is the inverse of an exponential function. The inverse of y = bx is y = logbx. Logarithm: The logarithm base b of a number x, logbx, is the power to which b must be raised to equal x. Natural exponential: Exponential expressions or functions with a base of e, i.e. y = e x . Natural logarithm: A logarithm with a base of e. A natural logarithm is the power, a, such that e a = b. The natural logarithm of x is written ln x. For example, ln 8 = 2.0794415… because e 2.0794415… = 8. Summarizing/Closing/Formative Assessment PROBLEM 2.Task 3: The Population of Exponentia Additional Practice Exercise. Inverse of Exponential Functions are Logarithmic Functions A Graph the inverse of exponential functions. B Graph logarithmic functions. We have seen in Math 2 that the inverse function of a quadratic function is the square root function. In this section we examine inverse functions of exponential functions, called logarithmic functions. A Graph the inverse of exponential functions. What does the inverse graph of the graph of the exponential function f(x) = 2 x look like? In the following example, we graph it using a t-table. Each point (x, y) on the graph of f(x) = 2 x is plotted as (y, x) on its inverse graph. Example 1. Graphing the inverse of an exponential function (i) Plot the inverse graph of f(x) = 2 x . (ii) Verify that a point (x, y) on f is plotted as (y, x) on its inverse graph. Solution (i) Create a t-table to graph y = 2 x . Next Interchanges x and y values in your t-table, and graph these points as this is the inverse graph, x = 2 y . Graph the line y = x. Graphing the three above gives the result shown in . x y = 2 x = y 2 Graph of Graph of -4.7 4.7 -3.1 3.1 Figure 1. (ii) shows the views when we toggle between the first two graphs (that is, trace to a point on one graph, then switch to the other graph). 2 3  8  log 2 8  3 exponent power In Math III, we'll evaluate logarithms. First, we graph logarithmic functions. Technical Note: Change of Base Formula Calculators have a built-in logarithmic function key, LOG, for the logarithm, base 10. To graph a logarithmic function having a base different than 10, we must use a change of base formula: log b x  log x log b . For example, to graph f x = log2x , we define y as log x ÷ log 2. Let’s first explore some graphs of logarithmic functions f x = logbx where b > 1. Example 2. Graphing f x =log b x , b > 1 Graph f x = logbx , for b = 1.5, 3, and 10. Describe any similarities and differences. Solution -1 -5 5 17.8 Figure 4. Graph of f x = log1.5x -1 -5 5 17.8 Figure 5. Graph of f x = log3x -1 -5 5 17.8 Figure 6. Graph of f x = log x By observing Figures 4,5, and 6, we see that for each graph, as x increases, so does y. The value log1.5x increases faster than the value log 3x , and log x has the slowest growth of the three. Since the three logarithmic functions are inverses of exponential functions y = 1. 5 x , y = 3 x , and y = 10 x , the point (0, 1) on the exponential curves is reflected as the point (1, 0) on each logarithmic graph. The horizontal asymptote of the exponential curves is the x-axis. It is reflected as the vertical asymptote, x = 0 (y-axis), on the logarithmic curves. The range of the exponential functions, 0,  , becomes the domain of each logarithmic function. The domain of the exponential functions is the set of real numbers, thus the range of each logarithmic function is the set of real numbers. The curves are all continuous and smooth. Below is a summary of the behavior of logarithmic functions whose base is greater than 1: f x = logbx , b > 1. 1. As x increases, so does y. 2. The y-axis is the vertical asymptote of the graph of f. 3. The graph passes through the point (1, 0). 4. The domain of f is 0,  , and the range is ,  . 5. The curve is continuous and smooth. Let’s look at some graphs of f x = logbx , where 0 < b < 1. Example 3. Graphing f x = logbx , 0 < b < 1 Graph f x = logbx , for b = 0.1, 1 3 , and 0.5. Describe any similarities and differences. Solution -1 -5 5 17.8 Figure 7. Graph of f x = log0 .1x -1 -5 5 17.8 Figure 8. Graph of f x = log 1 3 x -1 -5 5 17.8 Figure 9. Graph of f x = log0.5x By observing , , and , we see that for each graph, as x increases, y decreases. The value log 0.5x decreases faster than the value log 1 3 x , and log 0.1x has the slowest decrease of the three. The point (1, 0) is on each graph, and the y-axis is the asymptote. The domain of each function is 0,  ; the range is the set of real numbers. The three curves are all continuous and smooth.  Below is a summary of the behavior of logarithmic functions whose base is between 0 and 1: f x = logbx , 0 < b < 1. 1. As x increases, y decreases. 2. The y-axis is the asymptote of the graph of f. As x decreases to near 0,y increases without bound, that is, y increases to +. 3. The graph passes through the point (1, 0). 4. The domain of f is 0,  , and the range is ,  . 5. The curve is continuous and smooth. (Note that the only difference between the graphs of logarithmic functions f x = logbx when b > 1 and f x = logbx when 0 < b < 1 is that the former is an increasing function, and the latter is a decreasing function.) Exercises Math III Unit 3 A Graph the inverse of exponential functions. Exercises 1 - 2. Given the graph of f, sketch the graph of its inverse. 1. 4 4 -4 -4 2. 4-4 4 -4 Exercises 3 - 8. Plot the inverse graph of f. 3. f(x) = 4 x 5. f(x) = 10 x 5. f(x) = 3 x 6. f(x) = 5 x 7. f(x) = 0.3 x 8. f(x) = 0.7 x Exercises 9 - 10. For the given table of a function, switch rows to get a table that displays some points on the inverse graph. Plot the points displayed in each of the two tables and connect the points with a smooth curve. 9. x -2 -1 0 1 2 3 y 1 9 1 3 1 3 9 27 10. x -2 -1 0 1 2 3 y 1 16 1 4 1 4 16 64 Comment [ART1]: y = (1.5)^x Comment [ART2]: y = (1.5) ^ -x