Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 3 Exponential, Logistic, and Logarithmic Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.1 Exponential and Logistic Functions
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 What you’ll learn about Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models … and why Exponential and logistic functions model many growth patterns, including the growth of human and animal populations.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 Example Finding an Exponential Function from its Table of Values
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Growth and Decay
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Natural Base e
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Functions and the Base e
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Logistic Growth Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.2 Exponential and Logistic Modeling
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Population Model
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Growth and Decay Rates
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Growth and Decay Rates
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Bacteria Growth
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Bacteria Growth
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling a Rumor
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling a Rumor
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.3 Logarithmic Functions and Their Graphs
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Changing Between Logarithmic and Exponential Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverses of Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Basic Properties of Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide An Exponential Function and Its Inverse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Common Logarithm – Base 10 Logarithms with base 10 are called common logarithms. The common logarithm log 10 x = log x. The common logarithm is the inverse of the exponential function y = 10 x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Basic Properties of Common Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving Simple Logarithmic Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving Simple Logarithmic Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Basic Properties of Natural Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Graphs of the Common and Natural Logarithm
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Logarithmic Graphs
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Logarithmic Graphs
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Decibels
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.4 Properties of Logarithmic Functions
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Properties of Logarithms Change of Base Graphs of Logarithmic Functions with Base b Re-expressing Data … and why The applications of logarithms are based on their many special properties, so learn them well.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Properties of Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving the Product Rule for Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving the Product Rule for Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Expanding the Logarithm of a Product
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Expanding the Logarithm of a Product
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Condensing a Logarithmic Expression
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Condensing a Logarithmic Expression
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Change-of-Base Formula for Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Logarithms by Changing the Base
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Logarithms by Changing the Base
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.5 Equation Solving and Modeling
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression … and why The Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide One-to-One Properties
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving an Exponential Equation Algebraically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving an Exponential Equation Algebraically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a Logarithmic Equation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a Logarithmic Equation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: A kilometer is 3 orders of magnitude longer than a meter. A dollar is 2 orders of magnitude greater than a penny. New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Richter Scale
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H + ]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH=-log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Newton’s Law of Cooling
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + blnx Exponential regression:y = a·b x Power regression:y = a·x b
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Three Types of Logarithmic Re-Expression
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.6 Mathematics of Finance
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value … and why The mathematics of finance is the science of letting your money work for you – valuable information indeed!
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Interest Compounded Annually
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Compound Interest – Value of an Investment
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Annual Percentage Yield A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Future Value of an Annuity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Present Value of an Annuity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions