1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Logarithmic Functions and Models ♦Evaluate the common logarithm function ♦Solve basic exponential and logarithmic equations ♦Evaluate logarithms with other bases ♦Solve general exponential and logarithmic equations 5.4

2. Slide 5- 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Common Logarithm The common logarithm of a positive number x, denoted log x, is defined byThe common logarithm of a positive number x, denoted log x, is defined by logx = k if and only if x = 10 k logx = k if and only if x = 10 k where k is a real number. The function given by f(x) = log x is called the common logarithm function.The function given by f(x) = log x is called the common logarithm function.

3. Slide 5- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate each of the following. log10log10 log 100log 100 log 1000log 1000 log 10000log 10000 log (1/10)log (1/10) log (1/100)log (1/100) log (1/1000)log (1/1000) log 1log 1 1 because 10 1 = 101 because 10 1 = 10 2 because 10 2 = 1002 because 10 2 = 100 3 because 10 3 = 10003 because 10 3 = 1000 4 because 10 4 = 100004 because 10 4 = 10000 –1 because 10 -1 = 1/10–1 because 10 -1 = 1/10 –2 because 10 -2 = 1/100–2 because 10 -2 = 1/100 –3 because 10 -3 = 1/1000–3 because 10 -3 = 1/1000 0 because 10 0 = 10 because 10 0 = 1

4. Slide 5- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph of f(x) = log x xf(x)f(x).01-2.1 10 101 1002 Note that the graph of y = log x is the graph of y = 10 x reflected through the line y = x. This suggests that these are inverse functions.

5. Slide 5- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Inverse of y = log x Note that the graph of f(x) = log x passes the horizontal line test so it is a 1-1 function and has an inverse function.Note that the graph of f(x) = log x passes the horizontal line test so it is a 1-1 function and has an inverse function. Find the inverse of y = log xFind the inverse of y = log x Using the definition of common logarithm to solve for x givesUsing the definition of common logarithm to solve for x gives x = 10 y x = 10 y Interchanging x and y givesInterchanging x and y gives y = 10 x y = 10 x So yes, the inverse of y = log x is y = 10 xSo yes, the inverse of y = log x is y = 10 x

6. Slide 5- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inverse Properties of the Common Logarithm Recall that f -1 (x) = 10 x given f(x) = log x (f  f -1 )(x) = x for every x in the domain of f -1Since (f  f -1 )(x) = x for every x in the domain of f -1 log(log(10 x ) = x for all real numbers x. Since (f -1  f)(x) = x for every x in the domain of fSince (f -1  f)(x) = x for every x in the domain of f 10 logx = x for any positive number x10 logx = x for any positive number x

7. Slide 5- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Exponential Equations Using The Inverse Property log( Solving Exponential Equations Using The Inverse Property log(10 x ) = x Solve the equation 10 x = 35Solve the equation 10 x = 35

8. Slide 5- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Logarithmic Equations Using The Inverse Property 10 logx = x Solve the equation log x = 4.2Solve the equation log x = 4.2

9. Slide 5- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition of Logarithm With Base a The logarithm with base a of a positive number x, denoted by log a x is defined byThe logarithm with base a of a positive number x, denoted by log a x is defined by log a x = k if and only if x = a k where a > 0, a ≠1, and k is a real number. The function given by f(x) = log a x is called the logarithmic function with base a.The function given by f(x) = log a x is called the logarithmic function with base a.

10. Slide 5- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Practice with the Definition Log b c = d means____Log b c = d means____ p = log y m means ____p = log y m means ____ True or false:True or false: True or false: log 2 8 = 3True or false: log 2 8 = 3 True or false: log 5 25 = 2True or false: log 5 25 = 2 True or false: log 25 5 = 1/2True or false: log 25 5 = 1/2 True or false: log 4 8 = 2True or false: log 4 8 = 2 Practice Questions: What is the value of log 4 8?

11. Slide 5- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Practice Evaluating Logarithms log 6 36log 6 36 log 36 6log 36 6 log 2 32log 2 32 log 32 2log 32 2 log 6 (1/36)log 6 (1/36) log 2 (1/32)log 2 (1/32) log 100log 100 log (1/10)log (1/10) log 1log 1 Evaluate Answers:

12. Slide 5- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Calculators and logarithms The TI-83 evaluates base 10 logarithms and base e logarithms.The TI-83 evaluates base 10 logarithms and base e logarithms. Base 10 logs are called common logs.Base 10 logs are called common logs. log x means log 10 x.log x means log 10 x. Notice the log button on the calculator.Notice the log button on the calculator. Base e logs are called natural logs.Base e logs are called natural logs. ln x means log e x.ln x means log e x. Notice the ln button on the calculator.Notice the ln button on the calculator.

13. Slide 5- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate each of the following without calculator. Then check with calculator. lnelne ln(e 2 )ln(e 2 ) ln1ln1. lne = log e e = 1 since e 1 = elne = log e e = 1 since e 1 = e ln(e 2 ) = log e (e 2 ) = 2 since 2 is the exponent that goes on e to produce e 2.ln(e 2 ) = log e (e 2 ) = 2 since 2 is the exponent that goes on e to produce e 2. ln1 = log e 1 = 0 since e 0 = 1ln1 = log e 1 = 0 since e 0 = 1 1/2 since 1/2 is the exponent that goes on e to produce e 1/21/2 since 1/2 is the exponent that goes on e to produce e 1/2

14. Slide 5- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Inverse of y = log a x Note that the graph of f(x) = log a x passes the horizontal line test so it is a 1-1 function and has an inverse function.Note that the graph of f(x) = log a x passes the horizontal line test so it is a 1-1 function and has an inverse function. Find the inverse of y = log a xFind the inverse of y = log a x Using the definition of common logarithm to solve for x givesUsing the definition of common logarithm to solve for x gives x = a y x = a y Interchanging x and y givesInterchanging x and y gives y = a x y = a x So the inverse of y = log a x is y = a xSo the inverse of y = log a x is y = a x

15. Slide 5- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inverse Properties of Logarithms With Base a Recall that f -1 (x) = a x given f(x) = log a x (f  f -1 )(x) = x for every x in the domain of f -1Since (f  f -1 )(x) = x for every x in the domain of f -1 log a (log a (a x ) = x for all real numbers x. Since (f -1  f)(x) = x for every x in the domain of fSince (f -1  f)(x) = x for every x in the domain of f a log a x = x for any positive number xa log a x = x for any positive number x

16. Slide 5- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Exponential Equations Using The Inverse Property log a (a x ) = x Solve the equation 4 x = 1/64Solve the equation 4 x = 1/64

17. Slide 5- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Exponential Equations Using The Inverse Property log a (a x ) = x Solve the equation e x = 15Solve the equation e x = 15

18. Slide 5- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Logarithmic Equations Using The Inverse Property a log a x = x Solve the equation lnx = 1.5Solve the equation lnx = 1.5

19. Slide 5- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Recall from section 5.3 Graph of f(x) = a x where a >1 Graph of f(x) = a x where 0 < a <1 Using the fact that the graph of a function and its inverse are symmetric with respect to the line y = x, graph f -1 (x) = log a x for the two types of exponential functions listed above. Looking at the two resulting graphs, what is the domain of a logarithmic function? What is the range of a logarithmic function? Exponential Growth Function Exponential Decay Function

20. Slide 5- 20 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph of f(x) = a x where a >1 Graph of f(x) = a x where 0 < a < 1 Exponential Growth Function Exponential Decay Function Superimpose graphs of the inverses of the functions above similar to Figure 5.58 on page 422