Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and Logarithmic Equations 3.5 Exponential and Logarithmic Models 3.6 Exploring Data: Nonlinear Models
3.1 Exponential Functions and Their Graphs What You ll Learn: #50 - Recognize and evaluate exponential functions with base a. #51 - Graph exponential functions. #52 - Recognize, evaluate, and graph exponential functions with base e. #53 - Use exponential functions to model and solve real-life problems.
Definition of Exponential Function The exponential function f with base a is denoted by f x = a x where a > 0, a 1, and x is any real number.
Graphs of Exponential Functions
Exponential Function w/ a Negative Exponent
Properties of Exponents Page 177 1. a x a y = a x+y 2. ax = ay ax y 3. a x = 1 a x = 4. a 0 = 1 1 a x 5. ab x = a x b x 6. a x y = a xy 7. a b x = a x b x 8. a 2 = a 2 = a 2
Exploration Graph y = a x for each of the following: 1. Let a = 3,5, and 7 2. Let a = 1, 1, and 1 3 5 7
Application: Compound Interest For n compoundings per year: A = P 1 + r n nt For continuous compoundings per year: A = Pe rt
Finding the Balance for Compound Interest A total of $9,000 is invested at an annual interest rate of 2.5%, compounded annually. Find the balance in the account after 5 years. A = P 1 + r n nt Algebraic and Graphical Solution
Finding Compound Interest A total of $12,000 is invested at an annual interest rate of 3%. Find the balance after 4 years if the interest is compounded: a) Semi-Annually b) Quarterly c) Continuously
Radioactive Decay Let y represent a mass of radioactive strontium ( 90 Sr), in grams, whose half-life is 28 years. The quantity of strontium present after t years is y = 10 1 2 t 28. A. What is the initial mass (when t = 0)? B. How much of the initial mass is present after 80 years?
Population Growth The approximate number of fruit flies in an experimental population after t hours is given by Q t = 20e 0.03t, where t 0. A. Find the initial number of fruit flies in the population. B. How large is the population of fruit flies after 72 hours? C. Graph Q(t).
Homework Page 186 #55,56,59,64,65,68,70,77
3.2 Logarithmic Functions and Their Graphs What You ll Learn: #54 - Recognize and evaluate logarithmic functions with base a. #55 - Graph logarithmic functions. #56 - Recognize, evaluate, and graph natural logarithmic functions. #57 - Use logarithmic functions to model and solve real-life problems.
Does a x have an inverse? f x = a x Is it a one-to-one function? Does it pass the horizontal line test? YES! f x = a x f 1 x = log a x
Definition of Logarithmic Function For x > 0, a > 0, and a 1, y = log a x if and only if x = a y The function given by f x = log a x is read as log base a of x and is called the logarithmic function with base a
Exponential Logarithmic x = a y y = log a x
Evaluating Logarithms Use the definition of logarithmic functions to evaluate each function at the indicated value of x. A. f x = log 2 x, x = 32 B. f x = log 3 x, x = 1 C. f x = log 4 x, x = 2 D. f x = log 10 x, x = 1 100
The Common Logarithmic Function f x = log 10 x This is the log button on your calculator.
Use a calculator to evaluate the function f x = log 10 x at each value of x. A.x = 10 B.x = 2.5 C. x = 2 D.x = 1 4
Properties of Logarithms 1) log a 1 = 0 because a 0 = 1 2) log a a = 1 because a 1 = a 3) log a a x and a log a x = x 4) If log a x = log a y, then x = y
Using Properties of Logarithms Solve for x for each: 1. log 2 x = log 2 3 2. log 4 4 = x Simplify each: 1. log 5 5 x 2. 7 log 7 14
Graphs of Logarithmic Functions
The Natural Logarithmic Function For x > 0 y = ln x if and only if x = e y The function given by f x = log e x = ln x is called the natural logarithmic function.
Evaluating The Natural Logarithmic Function Use a calculator to evaluate the function f x = ln x at each indicated value of x. a) x = 2 b) x = 0.3 c) x = 1
Properties of Natural Logarithms 1) ln 1 = 0 because e 0 = 1 2) ln e = 1 because e 1 = e 3) ln e x = x because e ln x = x 4) If ln x = ln y, then x = y.
Using the properties of Natural Logs Use the properties of natural logs to rewrite each expression. 1. ln 1 e 2. e ln 5 3. ln e 0 4. 2 ln e
Domains of Log Functions Find the domain of each function: 1. f x = ln(x 2) 2. g x = ln(2 x) 3. h x = ln(x 2 )
Homework Page 195 #1-3,9-11,17,25-30,57-60, (73 a & b)
3.3 Properties of Logarithms What You ll Learn: #58 - Rewrite logarithms with different bases. #59 - Use properties of logarithms to evaluate or rewrite logarithmic expressions. #60 - Use properties of logarithms to expand or condense logarithmic expressions. #61 - Use logarithmic functions to model and solve real-life problems.
Change-of-Base Formula You can evaluate each of the following log expressions as such: log a x = log 10 x log 10 a or log a x = ln x ln a
Examples Changing Bases Evaluate the log expressions. 1. log 4 25 2. log 2 12
Properties of Logarithms 1. log a uv = log a u + log a v 2. log a ( u v ) = log a u log a v 3. log a u n = n log a u
CAUTION! log a u + v log a u + log a v ln u v ln u ln v
Using Properties of Logs Write each logarithm in terms of ln 2 and ln 3. 1. ln 6 2. ln 2 27
Rewriting Logarithmic Expressions Expand each expression: 1. log 4 5x 3 y 2. ln 3x 5 7 Condense each expression: 1. 1 2 log 10 x + 3 log 10 (x + 1) 2. 2 ln(x + 2) ln x
Homework Page 203 #9-13,17-20,23-32,45-48
3.4 Solving Exponential and Logarithmic Equations What You ll Learn: #62 - Solve simple exponential and logarithmic equations. #63 - Solve more complicated exponential equations. #64 - Solve more complicated logarithmic equations. #65 - Use exponential and logarithmic equations to model and solve real-life problems.
Examples Solving Logarithmic Equations 1. ln x = 2 2. log 3 (5x 1) = log 3 (x + 7) 3. 5 + 2 ln x = 4 4. 2 log 5 3x = 4
Examples Extraneous Solutions ln(x 2) + ln(2x 3) = 2 ln x
Examples Approximating the solution x = 2 x 2
Applications Page 212 Example 12 You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? How long to triple? A = Pe rt
Application Another Money Problem You are looking to deposit $200 into a savings account. The accounts you are interested in have monthly compounded interest. What interest rate should you seek if you want to: 1. Double your money in 15 years? 2. Reach $250 in 10 years?
Applications Page 212 Example 13 For selected year from 1980 to 2000, the average salary y (in thousands of dollars) for public school teachers for the year t can be modeled by the equation: y = 39. 2 + 23. 64 ln t, 10 t 30 where t = 10 represents 1980. During which year did the average salary for public school teachers reach $40.0 thousand?
Homework Page 213 Exponential #12, 18-60 MULT OF 3, 55 Logarithmic #73-89 ODD, 97, 98 Miscellaneous #103, 109, 111, 112
3.5 Exponential and Logarithmic Models What You ll Learn: #66 - Recognize the five most common types of models involving exponential or logarithmic functions. #67 - Use exponential growth and decay functions to model and solve real-life problems. #68 - Use Gaussian functions to model and solve real-life problems. #69 - Use logistic growth functions to model and solve real-life problems. #70 - Use logarithmic functions to model and solve real-life problems. *Handout*
Mathematical Models (5 of them) (that involve exp/log) 1)Exponential Growth 2)Exponential Decay 3)Gaussian 4)Logistic Growth y = 5)Logarithmic or y = ae bx, b > 0 y = ae bx, b > 0 y = ae x b 2 /c a 1+be rx y = a + b ln x y = a + b log 10 x
3.6 Exploring Data: Nonlinear Models What You ll Learn: #71 - Classify scatter plots. #72 - Use scatter plots and a graphing utility to find models for data and choose a model that best fits a set of data. #73 - Use a graphing utility to find exponential and logistic models for data. *Handout*
All Regression Programs (stat button over one click to calc) 1.Linear 2.Quadratic 3.Cubic 4.Quart 5.Natural Log 6.Exponential 7.Power 8.Logistic 9.Sin
Chapter 3 Review Page 239 #5-8, 29-40, 60, 66-68, 71-73, 80-85, 88-95, 98-105, 109-116