Introduction Exponential and logarithmic functions are commonly used to model real-life problems. They are particularly useful with situations in which the quantities being observed increase or decrease at a fast rate. A logarithmic function is the inverse of an exponential function; for the exponential function g(x) = 5 x, the inverse logarithmic function is x = log 5 g(x). By evaluating logarithms, patterns that lead to solutions can be observed. Graphing logarithmic functions can generally give you a more efficient way to see patterns in the variables. In this lesson, you will learn how to graph logarithmic functions : Graphs of Logarithmic Functions

Key Concepts The graph of an exponential function in the form f(x) = b x appears as a slight curve when b > 1, such as in the graph of f(x) = 2 x that follows : Graphs of Logarithmic Functions

Key Concepts, continued When b < 1, the graph resembles the graph of the positive value of b, but is reflected about the x-axis. For example, observe the graph of f(x) = –2 x, shown on the next slide. Notice that it is the reflection of the graph of f(x) = 2 x. (Note: When b = 1, the graph is a horizontal line.) : Graphs of Logarithmic Functions

Key Concepts, continued : Graphs of Logarithmic Functions

Key Concepts, continued The graph of an exponential function will approach an asymptote (a line that the graph of the function comes closer and closer to but never actually touches). In the graph of an exponential function, as the x-values increase, the y-values will either increase or decrease very rapidly. Logarithmic functions are inverses of exponential functions. Therefore, their graphs show similar behavior : Graphs of Logarithmic Functions

Key Concepts, continued The typical graph of a logarithmic function, f(x) = log x, is the mirror image of the exponential function. It has an asymptote as well. Furthermore, a logarithmic function also has a graph that shows a rapid increase or decrease in its y-values. For example, the graph on the next slide shows f(x) = log 2 x : Graphs of Logarithmic Functions

Key Concepts, continued : Graphs of Logarithmic Functions Notice that this graph has an asymptote at x = 0. For this graph, the domain (the set of x-values) is all real numbers greater than 0, while the range (the set of all y- values) is all real numbers.

Common Errors/Misconceptions incorrectly reading key features such as intercepts and asymptotes from a graph graphing a logarithmic function on a coordinate plane with a straight line instead of a curve incorrectly calculating asymptotes and intercepts : Graphs of Logarithmic Functions

Guided Practice Example 2 Three pairs of logarithmic functions are provided. Graph each pair on one coordinate plane. Then, use the graphs to identify and compare the asymptotes, intercepts, domains, and ranges for each pair, as well as the shape and end behavior of each pair of graphs. f(x) = –log x and g(x) = log x h(x) = –log 2 x and j(x) = log 2 x m(x) = –ln x and p(x) = ln x : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 1.Graph the first pair of functions on the same coordinate plane. Graph the given functions, f(x) = –log x and g(x) = log x, by hand or using a graphing calculator. The result should resemble the following : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 2.Compare the shapes and end behavior of the graphs. The shape and end behavior of the graph of f(x) = –log x are inverted from the shape and end behavior of the graph of g(x) = log x. In other words, the two graphs are reflections of each other. The line of reflection is the x-axis. The graph of f(x) = –log x decreases as x approaches positive infinity. The graph of g(x) = log x increases as x approaches positive infinity : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 3.Identify the two graphs’ asymptotes, intercepts, domains, and ranges. Since neither graph crosses the y-axis, both graphs have a vertical asymptote at x = 0. The graphs cross the x-axis at the x-intercept (1, 0). Since the x-values increase into positive infinity but never cross the y-axis, this means they are never negative. Thus, we can determine the domain of both graphs is x > 0. The range of both graphs is all real numbers : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 4.Graph the second pair of functions together on a new coordinate plane. The graph of j(x) = log 2 x and h(x) = –log 2 x is shown : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 5.Compare the shapes and end behavior of the second pair of graphs. The shape and end behavior of the graph of h(x) = –log 2 x are inverted from that of j(x) = log 2 x. In other words, each graph is a reflection of the other over the x-axis. The graph of h(x) = –log 2 x decreases as x approaches positive infinity. The graph of j(x) = log 2 x increases as x approaches positive infinity : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 6.Identify the two graphs’ asymptotes, intercepts, domains, and ranges. There is a vertical asymptote at x = 0 for each graph. Both graphs cross the x-axis at the x-intercept (1, 0). The domain of both graphs is x > 0. Since the y-values are not limited, we can determine the range of each graph is all real numbers : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 7.Graph the third pair of functions together on a new coordinate plane. Compare the graphs’ shapes and end behavior. The graph of p(x) = ln x and m(x) = –ln x is shown : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued The shape and end behavior of the graph of m(x) = –ln x are inverted from that of p(x) = ln x; therefore, the graphs are reflections of one another. The line of reflection is the x-axis. The graph of m(x) = –ln x decreases as x approaches positive infinity. The graph of p(x) = ln x increases as x approaches positive infinity : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 8.Use the graphs to identify the asymptotes, intercepts, domains, and ranges of this third pair of functions. There is a vertical asymptote at x = 0. The graphs cross the x-axis at the x-intercept (1, 0). The domain of both graphs is x > 0. The range of both graphs is all real numbers : Graphs of Logarithmic Functions

Guided Practice: Example 2, continued 9.Summarize your findings. In summary, all three pairs of graphs have the same vertical asymptote of x = 0. In every case, the x-intercept is (1, 0). The domain and range for all three pairs are identical. The three pairs of functions differ in the rate of increase or decrease as x increases : Graphs of Logarithmic Functions ✔

Guided Practice: Example 2, continued : Graphs of Logarithmic Functions

Guided Practice Example 3 Graph the logarithmic functions c(x) = log 2 (x + 1) and d(x) = log 2 x together on a coordinate plane. Use the graph to identify and compare the asymptotes, intercepts, domains, and ranges of the functions, as well as their shape and end behavior : Graphs of Logarithmic Functions

Guided Practice: Example 3, continued 1.Graph the functions together on a coordinate plane. Graph c(x) = log 2 (x + 1) and d(x) = log 2 x either by hand or using a graphing calculator. The result should resemble the following graph : Graphs of Logarithmic Functions

Guided Practice: Example 3, continued 2.Compare the shapes and end behavior of the graphs. The graph of c(x) = log 2 (x + 1) has the same shape and end behavior as d(x) = log 2 x, but the graph of c(x) is shifted to the left 1 unit. Both graphs increase as x approaches positive infinity : Graphs of Logarithmic Functions

Guided Practice: Example 3, continued 3.Identify the graphs’ asymptotes, intercepts, domains, and ranges. The graph of d(x) = log 2 x has a vertical asymptote at x = 0, whereas the graph of c(x) = log 2 (x + 1) has a vertical asymptote at x = –1. The graph of d(x) = log 2 x crosses the x-axis at the x-intercept (1, 0), whereas the graph of c(x) = log 2 (x + 1) crosses the x-axis at (0, 0). While the domain of d(x) = log 2 x is x > 0, the domain of c(x) = log 2 (x + 1) is x > –1. The range of both graphs is all real numbers : Graphs of Logarithmic Functions ✔

Guided Practice: Example 3, continued : Graphs of Logarithmic Functions