Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part I)

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 2.6 Logarithmic Functions  The student will be able to: Identify the graphs of one-to-one functions. Use and apply inverse functions. Evaluate logarithms. Rewrite log as exponential functions and vice versa.

3 Barnett/Ziegler/Byleen Business Calculus 12e One to One Functions

4 Barnett/Ziegler/Byleen Business Calculus 12e Which Functions Are One to One? One-to-one NOT One-to-one

5 Barnett/Ziegler/Byleen Business Calculus 12e Definition of Inverse Function

6 Finding the Inverse Function Barnett/Ziegler/Byleen Business Calculus 12e

7 Graphs of f and f -1 Barnett/Ziegler/Byleen Business Calculus 12e

8 Graphs of f and f -1 Barnett/Ziegler/Byleen Business Calculus 12e

9 Graphs of f and f -1 Barnett/Ziegler/Byleen Business Calculus 12e

10  Exponential functions are one-to-one because they pass the vertical and horizontal line tests. Barnett/Ziegler/Byleen Business Calculus 12e Logarithmic Functions

11 Barnett/Ziegler/Byleen Business Calculus 12e Inverse of an Exponential Function  Start with the exponential function:  Now, interchange x and y:  Solving for y:  The inverse of an exponential function is a log function.

12 Barnett/Ziegler/Byleen Business Calculus 12e Logarithmic Function The inverse of an exponential function is called a logarithmic function. For b > 0 and b  1,

13 Barnett/Ziegler/Byleen Business Calculus 12e Graphs

14 Transformations Barnett/Ziegler/Byleen Business Calculus 12e

15 Log Notation Barnett/Ziegler/Byleen Business Calculus 12e

16 Simple Logs Barnett/Ziegler/Byleen Business Calculus 12e

17 Log  Exponential  Think of the word “log” as meaning “exponent on base b”  To convert a log equation to an exponential equation: What’s the base? What’s the exponent? Write the equation  Barnett/Ziegler/Byleen Business Calculus 12e

18 Barnett/Ziegler/Byleen Business Calculus 12e Log  Exponential

19 Exponential  Log  To convert an exponential equation to a log equation: What’s the base? What’s the exponent? Write the equation  Check: Barnett/Ziegler/Byleen Business Calculus 12e

20 Barnett/Ziegler/Byleen Business Calculus 12e Exponential  Log

21 Barnett/Ziegler/Byleen Business Calculus 12e Solving Simple Equations

22 Using Your Calculator  Use your calculator to evaluate and round to 2 decimal places: Barnett/Ziegler/Byleen Business Calculus 12e

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Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part II)

25 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 2.6 Logarithmic Functions  The student will be able to: Use log properties. Solve log equations. Solve exponential equations.

26 Barnett/Ziegler/Byleen Business Calculus 12e Properties of Logarithms If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then

27 Using Properties Barnett/Ziegler/Byleen Business Calculus 12e

28 Barnett/Ziegler/Byleen Business Calculus 12e Solving Log Equations Solve for x: x can’t be -10 because you can’t take the log of a negative number.

29 Barnett/Ziegler/Byleen Business Calculus 12e Solving Log Equations Solve for x. Obtain the exact solution of this equation in terms of e. ln (x + 1) – ln x = 1 ex = x + 1 ex - x = 1 x(e - 1) = 1

30 Solving Exponential Equations  Method 1: Convert the exponential equation to a log equation. Then evaluate. Barnett/Ziegler/Byleen Business Calculus 12e

31 Solving Exponential Equations  Method 2: Isolate the exponential part on one side, then take the log or ln of both sides of the equation. Then evaluate. Barnett/Ziegler/Byleen Business Calculus 12e

32 Solving Exponential Equations  Solve and round answer to 4 decimal places: Barnett/Ziegler/Byleen Business Calculus 12e

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Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part III)

35 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 2.6 Logarithmic Functions  The student will be able to: Solve applications involving logarithms.

36 Barnett/Ziegler/Byleen Business Calculus 12e Application: Finance How long will it take money to double if compounded monthly at 4% interest? You can take the log or the ln of both sides. It will take about 17.4 yrs for the money to double.

37 Barnett/Ziegler/Byleen Business Calculus 12e Application: Finance  Suppose you invest $1500 into an account that is compounded continuously. At the end of 10 years, you want to have a balance of $6500. What must the annual percentage rate be? The annual percentage rate must be 14.7%

38 Barnett/Ziegler/Byleen Business Calculus 12e Application: Archeology The fossil would be 15,299 years old.

39 Application: Sound Intensity Barnett/Ziegler/Byleen Business Calculus 12e

40 Application: Sound Intensity  Solve for N: Barnett/Ziegler/Byleen Business Calculus 12e

41 Application: Sound Intensity Barnett/Ziegler/Byleen Business Calculus 12e The sound of heavy traffic is about 80 decibels.

42 Barnett/Ziegler/Byleen Business Calculus 12e Logarithmic Regression When the scatter plot of a data set indicates a slowly increasing or decreasing function, a logarithmic function often provides a good model. We use logarithmic regression on a graphing calculator to find the function of the form y = a + b*ln(x) that best fits the data.

43 Barnett/Ziegler/Byleen Business Calculus 12e Example of Logarithmic Regression A cordless screwdriver is sold through a national chain of discount stores. A marketing company established the following price-demand table, where x is the number of screwdrivers in demand each month at a price of p dollars per screwdriver. x p = D(x) 1, , , , ,00053 Find a log regression equation to predict the price per screwdriver if the demand reaches 6,000.

44 Barnett/Ziegler/Byleen Business Calculus 12e Example of Logarithmic Regression x p = D(x) 1, , , , ,00053

45 Barnett/Ziegler/Byleen Business Calculus 12e Example of Logarithmic Regression Xmax=6500 Trace Up arrow Enter 6000

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