Logarithmic Functions and Their Graphs

Transcription

1 Logarithmic Functions and Their Graphs Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 1 / 24

2 Logarithmic Functions and Their Graphs 1 Logarithmic Functions Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 2 / 24

3 Logarithmic Functions and Their Graphs 1 Logarithmic Functions 2 Properties of Logarithms Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 2 / 24

4 Logarithmic Functions and Their Graphs 1 Logarithmic Functions 2 Properties of Logarithms 3 Graphs of Logarithmic Functions Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 2 / 24

5 Logarithmic Functions and Their Graphs 1 Logarithmic Functions 2 Properties of Logarithms 3 Graphs of Logarithmic Functions 4 Natural Logarithms Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 2 / 24

6 Logarithmic Functions Back at the end of Chapter 1, you were taught that if a function is one-to-one, then the function must have an inverse function. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 3 / 24

7 Logarithmic Functions Back at the end of Chapter 1, you were taught that if a function is one-to-one, then the function must have an inverse function. In the last section we discussed functions of the form f (x) = a x and we determined that it passes the Horizontal Line Test, so it is also one-to-one. Thus, it also must have an inverse function. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 3 / 24

8 Logarithmic Functions Back at the end of Chapter 1, you were taught that if a function is one-to-one, then the function must have an inverse function. In the last section we discussed functions of the form f (x) = a x and we determined that it passes the Horizontal Line Test, so it is also one-to-one. Thus, it also must have an inverse function. The inverse function is called the logarithmic function with base a. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 3 / 24

9 Logarithmic Functions Back at the end of Chapter 1, you were taught that if a function is one-to-one, then the function must have an inverse function. In the last section we discussed functions of the form f (x) = a x and we determined that it passes the Horizontal Line Test, so it is also one-to-one. Thus, it also must have an inverse function. The inverse function is called the logarithmic function with base a. Logarithmic Function with Base a For x > 0, a > 0, and a 1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 3 / 24

10 Equivalent Forms of the Logarithmic Function The equations are equivalent. y = log a x and x = a y Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 4 / 24

11 Equivalent Forms of the Logarithmic Function The equations y = log a x and x = a y are equivalent. The first equation is in logarithmic form and the second equation is in exponential form. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 4 / 24

12 Evaluating Logarithms Practice Use the definition of logarithmic function and evaluate each logarithm at the indicated value of x. a f (x) = log 2 x, x = 32 b f (x) = log 3 x, x = 1 c f (x) = log 4 x, x = 2 d f (x) = log 10 x, x = Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 5 / 24

13 Common Logarithmic Function The logarithmic function with base 10 is called the common logarithmic function. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 6 / 24

14 Common Logarithmic Function The logarithmic function with base 10 is called the common logarithmic function. It can technically be written as log 10, but is more commonly written more simply as log. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 6 / 24

15 Evaluating Common Logarithms on a Calculator Practice Use a calculator to evaluate the function given by f (x) = log x at each value of x. a x = 10 b x = 1 3 c x = 2.5 d x = 2 Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 7 / 24

16 Properties of Logarithms Properties of Logarithms The following properites follow directly form the definition of the logarithmic function with base a. 1 log a 1 = 0 because a 0 = 1. 2 log a a = 1 because a 1 = 0. 3 log a a x = x and a log a x = x. 4 If log a x = log a y, then x = y. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 8 / 24

17 Using Properties of Logarithms Example Simplify the following logarithms. a log 4 1 b log 7 7 c 6 log 6 20 Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 9 / 24

18 Using Properties of Logarithms Practice Simplify the following logarithms. a log 4 1 b log 7 7 c 6 log 6 20 Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 10 / 24

19 Extraneous Solutions When you utilize the one-to-one property to solve logarithmic equations, it is possible that you may end up with an extraneous solution. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 11 / 24

20 Extraneous Solutions When you utilize the one-to-one property to solve logarithmic equations, it is possible that you may end up with an extraneous solution. Extraneous solutions are solutions that the algebra yields, but are not actually solutions to the original equation. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 11 / 24

21 Extraneous Solutions When you utilize the one-to-one property to solve logarithmic equations, it is possible that you may end up with an extraneous solution. Extraneous solutions are solutions that the algebra yields, but are not actually solutions to the original equation. Each time you solve a logarithmic equation, you will need to check to make sure that your solution is not extraneous. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 11 / 24

22 Using the One-to-One Property Example Solve the equation log 3 x = log 3 12 for x. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 12 / 24

23 Using the One-to-One Property Practice Solve the following equations. a log(2x + 1) = log x b log 4 (x 2 6) = log 4 10 Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 13 / 24

24 Logarithmic Functions and Their Graphs (Part 1 of 3) Assignment pg #2-22 even, Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 14 / 24

25 Graphs of Logarithmic Functions Recall that on a graph, you can determine if two functions are inverses if the graphs are reflections of each other over the line y = x. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 15 / 24

26 Graphs of Logarithmic Functions Recall that on a graph, you can determine if two functions are inverses if the graphs are reflections of each other over the line y = x. To sketch the graph of y = log a x, you can use the fact that the graphs of inverse functions are reflections since we can typically graph y = a x without a calculator. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 15 / 24

27 Graphs of Exponential and Logarithmic Functions Example In the same coordinate plane, sketch the graph of f (x) = 2 x and g(x) = log 2 x Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 16 / 24

28 Transformations of Graphs of Logarithmic Functions Calculator Demonstration Graph f (x) = log x. Then graph each of the following and describe the translations necessary to graph each from the parent function f (x) = 3 x. a g(x) = log(x 1) b h(x) = 2 + log x Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 17 / 24

29 Logarithmic Functions and Their Graphs (Part 2 of 3) Assignment pg #2-22 even, 27-30, odd, Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 18 / 24

30 The Natural Logarithmic Function The Natural Logarithmic Function The function defined by f (x) = log e x = ln x, x > 0 is called the natural logarithmic function. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 19 / 24

31 The Natural Logarithmic Function The Natural Logarithmic Function The function defined by f (x) = log e x = ln x, x > 0 is called the natural logarithmic function. The natural logarithmic function is read as the natural log of x or el en of x. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 19 / 24

32 The Natural Logarithmic Function The Natural Logarithmic Function The function defined by f (x) = log e x = ln x, x > 0 is called the natural logarithmic function. The natural logarithmic function is read as the natural log of x or el en of x. Notice that the natural logarithm is written without a base. The base should be understood to be e. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 19 / 24

33 Equivalent Forms of Natural Logarithmic Functions Every logarithmic equation can be written in an equivalent exponential form and every exponential equation can be written in logarithmic form. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 20 / 24

34 Equivalent Forms of Natural Logarithmic Functions Every logarithmic equation can be written in an equivalent exponential form and every exponential equation can be written in logarithmic form. This means that y = ln x and x = e y are equivalent functions. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 20 / 24

35 Equivalent Forms of Natural Logarithmic Functions Every logarithmic equation can be written in an equivalent exponential form and every exponential equation can be written in logarithmic form. This means that y = ln x and x = e y are equivalent functions. Additionally, you know then that the graphs of f (x) = e x and g(x) = ln x are reflections over the line y = x since they are inverses of one another. Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 20 / 24

36 Properties of Natural Logarithms Properties of Natural Logarithms 1 ln 1 = 0 because e 0 = 1 2 ln e = 1 because e 1 = e 3 ln e x = x and e ln x = x 4 If ln x = ln y, then x = y Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 21 / 24

37 Using Properties of Natural Logarithms Practice Use the properties of natural logarithms to simplify each expression. a ln 1 e b e ln 5 c ln 1 3 d 2 ln e Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 22 / 24

38 Finding the Domains of Logarithmic Functions Practice Find the domain of each function. Use interval notation. a f (x) = ln(x 2) b g(x) = ln(2 x) c h(x) = ln x 2 Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 23 / 24

39 Logarithmic Functions and Their Graphs (Part 3 of 3) Assignment pg #2-22 even, 27-30, odd, 39-44, even, 69-72, odd Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 24 / 24