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![©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 3 – Writing Logarithms in Terms of
Simpler Logarithms
a.
b.
wzx
wzx
zwx
zw
x
lnlnln
)ln(lnln
)ln(lnln
−−=
+−=
−=
)]3ln()2ln(8ln5[
3
1
)]3ln()2ln([ln
3
1
)}3ln(])2({ln[
3
1
3
)2(
ln
3
1
3
)2(
ln
3
)2(
ln
85
85
85
3/1
85
3
85
−−−+=
−−−+=
−−−=
−
−
=
−
−
=
−
−
xxx
xxx
xxx
x
xx
x
xx
x
xx](https://image.slidesharecdn.com/chapter4-exponentialandlogarithmicfunctions-151003150209-lva1-app6891/75/Chapter-4-Exponential-and-Logarithmic-Functions-19-2048.jpg)
Uploaded byMuhammad Bilal Khairuddin
Chapter 4 - Exponential and Logarithmic Functions
This chapter discusses exponential and logarithmic functions. It begins by introducing exponential functions and their properties, including examples of exponential growth and decay. Logarithmic functions are then introduced as the inverse of exponential functions. The chapter covers the graphs and properties of logarithmic functions, including logarithmic equations and how to solve them. Examples are provided to demonstrate solving exponential and logarithmic equations that arise in business, economics, and science applications.
In this document
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Overview of the chapter on Exponential and Logarithmic Functions, covering key sections in Mathematical Analysis.
The chapter aims to introduce exponential functions, logarithmic functions, and solving related equations.
Chapter content includes sections on Exponential Functions, Logarithmic Functions, and their properties.
Defines exponential functions with base b, explores bacteria growth, graphs, and compound interest examples.
Examples involving population growth, projections, and radioactive decay showcasing practical applications of exponential models.
Definitions and conversions from exponential to logarithmic forms, along with specific examples and graphs.
Solving problems related to logarithms including half-life calculations, properties, and simplifications.
Differentiating between logarithmic and exponential equations, examples like oxygen consumption and predator-prey models.
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- 4. ©2007 Pearson EducationAsia • To introduce exponential functions and their applications. • To introduce logarithmic functions and their graphs. • To study the basic properties of logarithmic functions. • To develop techniques for solving logarithmic and exponential equations. Chapter 4: Exponential and Logarithmic Functions Chapter ObjectivesChapter Objectives
- 5. ©2007 Pearson EducationAsia Exponential Functions Logarithmic Functions Properties of Logarithms Logarithmic and Exponential Equations 4.1) 4.2) 4.3) 4.4) Chapter 4: Exponential and Logarithmic Functions Chapter OutlineChapter Outline
- 6. ©2007 Pearson EducationAsia • The function f defined by where b > 0, b ≠ 1, and the exponent x is any real number, is called an exponential function with base b1 . Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions4.1 Exponential Functions ( ) x bxf =
- 7. ©2007 Pearson EducationAsia The number of bacteria present in a culture after t minutes is given by . a. How many bacteria are present initially? b. Approximately how many bacteria are present after 3 minutes? Solution: a. When t = 0, b. When t = 3, Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Example 1 – Bacteria Growth ( ) t tN = 3 4 200 0 4 (0) 300 300(1) 300 3 N = = = ÷ 3 4 64 6400 (3) 300 300 711 3 27 9 N = = = ≈ ÷ ÷
- 8. ©2007 Pearson EducationAsia Graph the exponential function f(x) = (1/2)x . Solution: Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Example 3 – Graphing Exponential Functions with 0 < b < 1
- 9. ©2007 Pearson EducationAsia Properties of Exponential Functions Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions
- 10. ©2007 Pearson EducationAsia Solution: Compound Interest • The compound amount S of the principal P at the end of n years at the rate of r compounded annually is given by . Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Example 5 – Graph of a Function with a Constant Base 2 Graph 3 .x y = (1 )n S P r= +
- 11. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Example 7 – Population Growth The population of a town of 10,000 grows at the rate of 2% per year. Find the population three years from now. Solution: For t = 3, we have .3 (3) 10,000(1.02) 10,612P = ≈
- 12. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Example 9 – Population Growth The projected population P of a city is given by where t is the number of years after 1990. Predict the population for the year 2010. Solution: For t = 20, 0.05(20) 1 100,000 100,000 100,000 271,828P e e e= = = ≈ 0.05 100,000 t P e=
- 13. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Example 11 – Radioactive Decay A radioactive element decays such that after t days the number of milligrams present is given by . a. How many milligrams are initially present? Solution: For t = 0, . b. How many milligrams are present after 10 days? Solution: For t = 10, . 0.062 100 t N e− = ( ) mg100100 0062.0 == − eN ( ) mg8.53100 10062.0 == − eN
- 14. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.2 Logarithmic Functions4.2 Logarithmic Functions Example 1 – Converting from Exponential to Logarithmic Form • y = logbx if and only if by =x. • Fundamental equations are and logb x b x=log x b b x= 2 5 4 a. Since 5 25 it follows that log 25 2 b. Since 3 81 it follo Exponential Form Logarithmic Form = = = 3 0 10 ws that log 81 4 c. Since 10 1 it follows that log 1 0 = = =
- 15. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.2 Logarithmic Functions Example 3 – Graph of a Logarithmic Function with b > 1 Sketch the graph of y = log2x. Solution:
- 16. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.2 Logarithmic Functions Example 5 – Finding Logarithms a. Find log 100. b. Find ln 1. c. Find log 0.1. d. Find ln e-1 . d. Find log366. ( ) 210log100log 2 == 01ln = 110log1.0log 1 −== − 1ln1ln 1 −=−=− ee 2 1 6log2 6log 6log36 ==
- 17. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.2 Logarithmic Functions Example 7 – Finding Half-Life • If a radioactive element has decay constant λ, the half-life of the element is given by A 10-milligram sample of radioactive polonium 210 (which is denoted 210 Po) decays according to the equation. Determine the half-life of 210 Po. Solution: λ 2ln =T days λ T 4.138 00501.0 2ln2ln ≈==
- 18. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.3 Properties of Logarithms4.3 Properties of Logarithms Example 1 – Finding Logarithms • Properties of logarithms are: nmmn bbb loglog)(log.1 += nm n m bb logloglog.2 b −= mrm b r b loglog3. = a. b. c. d. 7482.18451.09031.07log8log)78log(56log =+≈+=⋅= 6532.03010.09542.02log9log 2 9 log =−≈−= 8062.1)9031.0(28log28log64log 2 =≈== 3495.0)6990.0( 2 1 5log 2 1 5log5log 2/1 =≈== b m m b m m a a b b b bb log log log.7 1log.6 01log.5 log 1 log4. = = = −=
- 19. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.3 Properties of Logarithms Example 3 – Writing Logarithms in Terms of Simpler Logarithms a. b. wzx wzx zwx zw x lnlnln )ln(lnln )ln(lnln −−= +−= −= )]3ln()2ln(8ln5[ 3 1 )]3ln()2ln([ln 3 1 )}3ln(])2({ln[ 3 1 3 )2( ln 3 1 3 )2( ln 3 )2( ln 85 85 85 3/1 85 3 85 −−−+= −−−+= −−−= − − = − − = − − xxx xxx xxx x xx x xx x xx
- 20. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.3 Properties of Logarithms Example 5 – Simplifying Logarithmic Expressions a. b. c. d. e. .3ln 3 xe x = 3 30 10log01000log1log 3 = += +=+ 9 89/8 7 9 8 7 7log7log == 1)3(log 3 3 log 81 27 log 1 34 3 33 −== = − 0)1(1 10logln 10 1 logln 1 =−+= +=+ − ee
- 21. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.3 Properties of Logarithms Example 7 – Evaluating a Logarithm Base 5 Find log52. Solution: 4307.0 5log 2log 2log5log 2log5log 25 ≈= = = = x x x x
- 22. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.3 Properties of Logarithms 4.4 Logarithmic and Exponential Equations4.4 Logarithmic and Exponential Equations • A logarithmic equation involves the logarithm of an expression containing an unknown. • An exponential equation has the unknown appearing in an exponent.
- 23. ©2007 Pearson EducationAsia An experiment was conducted with a particular type of small animal. The logarithm of the amount of oxygen consumed per hour was determined for a number of the animals and was plotted against the logarithms of the weights of the animals. It was found that where y is the number of microliters of oxygen consumed per hour and x is the weight of the animal (in grams). Solve for y. Chapter 4: Exponential and Logarithmic Functions 4.4 Logarithmic and Exponential Equations Example 1 – Oxygen Composition xy log885.0934.5loglog +=
- 24. ©2007 Pearson EducationAsia Solution: Chapter 4: Exponential and Logarithmic Functions 4.4 Logarithmic and Exponential Equations Example 1 – Oxygen Composition )934.5log(log log934.5log log885.0934.5loglog 885.0 885.0 xy x xy = += += 885.0 934.5 xy =
- 25. ©2007 Pearson EducationAsia Chapter 4: Exponential and Logarithmic Functions 4.4 Logarithmic and Exponential Equations Example 3 – Using Logarithms to Solve an Exponential Equation Solution: .124)3(5 1 =+ −x Solve 61120.1 ln4ln 4 124)3(5 3 71 3 71 1 ≈ = = =+ − − − x x x x
- 26. ©2007 Pearson EducationAsia In an article concerning predators and prey, Holling refers to an equation of the form where x is the prey density, y is the number of prey attacked, and K and a are constants. Verify his claim that Solution: Find ax first, and thus Chapter 4: Exponential and Logarithmic Functions 4.4 Logarithmic and Exponential Equations Example 5 – Predator-Prey Relation ax yK K = − ln )1( ax eKy − −= K yK e e K y eKy ax ax ax − = −= −= − − − 1 )1( ax yK K ax K yK ax K yK = − = − − −= − ln ln ln (Proved!)