Slide 3- 1

Chapter 3 Exponential, Logistic, and Logarithmic Functions

3.1 Exponential and Logistic Functions

Slide 3- 4 Quick Review

Slide 3- 5 Quick Review Solutions

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.

Slide 3- 7 Exponential Functions

Slide 3- 8 Exponential Functions Rules For Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

Slide 3- 9 Use the rules for exponents to solve for x 4 x = 128 (2) 2x = 2 7 2x = 7 x = 7/2 2 x = 1/32 2 x = 2 -5 x = -5 Exponential Functions

Slide (x 3 y 2/3 ) 1/2 x 3/2 y 1/3 27 x = 9 -x+1 (3 3 ) x = (3 2 ) -x+1 3 3x = 3 -2x+2 3x = -2x+ 2 5x = 2 x = 2/5 Exponential Functions

Slide Example Finding an Exponential Function from its Table of Values

Slide Example Finding an Exponential Function from its Table of Values

Slide y x y = 2 x If b > 1, then the graph of b x will: Rise from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

Slide y x y = ( 1 / 2 ) x If 0 < b < 1, then the graph of b x will: Fall from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

Slide Example Transforming Exponential Functions Describe how to transform the graph of f(x) = 2 x into the graph g(x) = 2 x-2 The graph of g(x) = 2 x-2 is obtained by translat ing the graph of f(x) = 2 x by 2 units to the right.

Slide Example Transforming Exponential Functions

Slide Example Transforming Exponential Functions

Slide The Natural Base e

Slide Exponential Functions and the Base e

Slide Exponential Functions and the Base e

Slide Example Transforming Exponential Functions

Slide Example Transforming Exponential Functions

Slide Logistic Growth Functions

Slide Exponential Growth and Decay

Slide Exponential Functions Definitions Exponential Growth and Decay The function y = k a x, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. y new amount y O original amount b base t time h half life

Slide Exponential Functions An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. (a)Find the amount remaining after t hours. (b)Find the amount remaining after 60 hours. a. y = y o b t/h y = 2 (1/2) (t/15) b. y = y o b t/h y = 2 (1/2) (60/15) y = 2(1/2) 4 y =.125 g

Slide Exponential Functions A bacteria double every three days. There are 50 bacteria initially present (a)Find the amount after 2 weeks. (b)When will there be 3000 bacteria? a. y = y o b t/h y = 50 (2) (14/3) y = 1269 bacteria

Slide Exponential Functions A bacteria double every three days. There are 50 bacteria initially present When will there be 3000 bacteria? b. y = y o b t/h 3000 = 50 (2) (t/3) 60 = 2 t/3

3.2 Exponential and Logistic Modeling

Slide Quick Review

Slide Quick Review Solutions

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.

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.

Slide Exponential Population Model

Slide Example Finding Growth and Decay Rates

Slide Example Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.

Slide Example Modeling Bacteria Growth

Slide Example Modeling Bacteria Growth

Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.

Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.

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.

Slide Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor which spreads logistically so that s(t) = 1500/( e.-.09t ) models the number of students who have heard the rumor at the end of t days, where t = 0 is the day the rumor begins to spread (a)How many students have heard the rumor by the end of Day 0? (b)How long does it take for 1000 students to hear the rumor?

Slide Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor which spreads logistically so that s(t) = 1500/( e.-.09t ) models the number of students who have heard the rumor at the end of t days, where t = 0 is the day the rumor begins to spread (a)How many students have heard the rumor by the end of Day 0? (b)How long does it take for 1000 students to hear the rumor?

3.3 Logarithmic Functions and Their Graphs

Slide Quick Review

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.

Slide Logarithmic Functions The inverse of an exponential function is called a logarithmic function. Definition: x = a y if and only if y = log a x

Slide Changing Between Logarithmic and Exponential Form

Slide Logarithmic Functions log 4 16 = 2 ↔ 4 2 = 16 log 3 81 = 4 ↔ 3 4 = 81 log = 2 ↔ 10 2 = 100

Slide Inverses of Exponential Functions

Slide Logarithmic Functions The function f (x) = log a x is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote: x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)

Slide Basic Properties of Logarithms

Slide An Exponential Function and Its Inverse

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.

Slide Basic Properties of Common Logarithms

Slide Example Solving Simple Logarithmic Equations

Slide Basic Properties of Natural Logarithms

Slide Graphs of the Common and Natural Logarithm

Slide Example Transforming Logarithmic Graphs

Slide Example Transforming Logarithmic Graphs

Slide Decibels

3.4 Properties of Logarithmic Functions

Slide Quick Review

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.

Slide log a (a x ) = x for all x   2. a log a x = x for all x > 0 3. log a (xy) = log a x + log a y 4. log a (x/y) = log a x – log a y 5. log a x n = n log a x Common Logarithm: log 10 x = log x Natural Logarithm: log e x = ln x All the above properties hold. Logarithmic Functions

Slide Properties of Logarithms

Slide Example Proving the Product Rule for Logarithms

Slide Example Proving the Product Rule for Logarithms

Slide Example Expanding the Logarithm of a Product

Slide Example Expanding the Logarithm of a Product

Slide Example Condensing a Logarithmic Expression

Slide Example Condensing a Logarithmic Expression

Slide Logarithmic Functions Product Rule

Slide Quotient Rule Logarithmic Functions

Slide Power Rule Logarithmic Functions

Slide Expand Logarithmic Functions

Slide Change-of-Base Formula for Logarithms

Slide Example Evaluating Logarithms by Changing the Base

Slide Example Evaluating Logarithms by Changing the Base

3.5 Equation Solving and Modeling

Slide Quick Review

Slide Quick Review Solutions

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.

Slide One-to-One Properties

Slide Example Solving an Exponential Equation Algebraically

Slide Example Solving an Exponential Equation Algebraically

Slide Example Solving a Logarithmic Equation

Slide Example Solving a Logarithmic Equation

Slide Solving Exponential Equations To solve exponential equations, pick a convenient base (often base 10 or base e) and take the log of both sides. Solve:

Slide Take the log of both sides: Power rule: Solving Exponential Equations

Slide Solve for x: Divide: Solving Exponential Equations

Slide To solve logarithmic equations, write both sides of the equation as a single log with the same base, then equate the arguments of the log expressions. Solve: Solving Exponential Equations

Slide Write the left side as a single logarithm: Solving Exponential Equations

Slide Equate the arguments: Solving Exponential Equations

Slide Solve for x: Solving Exponential Equations

Slide Solving Exponential Equations

Slide Check for extraneous solutions. x = -3, since the argument of a log cannot be negative Solving Exponential Equations

Slide To solve logarithmic equations with one side of the equation equal to a constant, change the equation to an exponential equation Solve: Solving Exponential Equations

Slide Write the left side as a single logarithm: Solving Exponential Equations

Slide Write as an exponential equation: Solving Exponential Equations

Slide Solve for x: Solving Exponential Equations

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.

Slide Richter Scale

Slide Graphs of Logarithmic Functions What is the magnitude on the Richter scale of an earthquake if a = 300, T = 30 and B = 1.2?

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.

Slide Newton’s Law of Cooling

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?

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?

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

Slide Three Types of Logarithmic Re-Expression

Slide Three Types of Logarithmic Re-Expression (cont’d)

Slide Three Types of Logarithmic Re-Expression (cont’d)

3.6 Mathematics of Finance

Slide Quick Review

Slide Quick Review Solutions

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!

Slide Interest Compounded Annually

Slide Interest Compounded k Times per Year

Slide Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

Slide Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

Slide Compound Interest – Value of an Investment

Slide Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

Slide Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

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.

Slide Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

Slide Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

Slide Future Value of an Annuity

Slide Present Value of an Annuity

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test Solutions

Slide Chapter Test Solutions

Slide Chapter Test Solutions