Chapter 4: 4.1: Exponential Functions Definition: Graphs of y = b x Exponential and Logarithmic Functions The Exponential function f with base b is f (x) = b x where b > 0, b 1, x a real number Graph: f (x) = 2 x and g(x) = 4 x x f (x) = 2 x -2-1 0 1 2 4 g(x) = 4 x Now graph h(x) = 2 -x x h(x) -2-1 0 1 2 Characteristics of Exponential Functions of form f (x) = b x (p.415) 1) The domain consists of all R. The range is all R +. 2) All graphs have the point (0, 1) in common. 3) If b > 1, the graph is one of exponential growth. 4) If 0 < b <1, the graph is one of exponential decay. 5) The function f (x) = b x is 1-1 (one-to-one). 6) The x-axis (or the line y = 0) is a horizontal asymptote. From J. P. Wood, Spring 2012 1
Sketching Graphs of Exponentials Look at y = 3 x [-2, 2]1, [-2, 3]1 Now g(x) = 3 x+1 Now h(x) = 3 x 2 Now k(x) = -3 x Now t(x) = 3 -x Transformations of Exponential Functions (p.416) for f (x) = b x Vertical: g (x) = b x + c g (x) = b x c Horizontal: Reflection: g (x) = b x+c g (x) = b x-c g (x) = -b x g (x) = b -x Vertical Stretch/Shrink: g (x) = c b x Horizontal Stretch/Shrink: g (x) = b c x From J. P. Wood, Spring 2012 2
The Natural Number e Let s look at the expression: (1 + ) n Evaluate for n = 1, 10, 10 2, 10 3, 10 6 1 n Graph: f (x) = e x Try g (x) = 2e x k (x) = 4e x t (x) = ½e x l (x) = e 2x m (x) = e 4x w (x) = e ½x From J. P. Wood, Spring 2012 3
Formulas for Compound Interest After t years, the balance A in an account with principal P and an annual interest rate r (in decimal form) is given by: 1. For n compounding per year: A = P( 1 + ) nt 2. For continuous compounding: A = Pe r t Ex, A total of $12,000 is invested at an annual rate of 4%. Find the balance after 5 years if it is compounded : a) Quarterly r n b) Continuously Ex, You have a choice of investing $8000 over 6 years at 7% compounded quarterly or 6.95% compounded continuously. Which is the better investment? From J. P. Wood, Spring 2012 4
4.2: Logarithmic Functions Definition: Logarithmic Function For x > 0 and b > 0, b 1 y = log b x if and only if b y = x The function f (x) = log b x is called the logarithmic function with base b. For r > 0 and 0 < b 1, log b r = t b t = r Properties of Logarithms 1. log b 1 = 0 (because b 0 = 1) 2. log b b = 1 (because b 1 = b) 3. log b b x = x (because b x = b x ) 4. b log b x = x (because log b x= log b x) Graphs of Exponential and Logarithmic Functions From J. P. Wood, Spring 2012 5
Characteristics of Logarithmic Functions of form f (x) = log b x 1) The domain consists of all R +. The range all R. 2) All graphs have the point (1, 0) in common. 3) If b > 1, the graph goes up and to the right. 4) If 0 < b <1, the graph goes down and to the right. 5) The graph approaches but does not touch the y-axis Graph: f (x) = log 10 x with calculator [-1,10]1,[-2,2]1 f (x) = log 10 x = f (x) log x Sketch: g (x) = log (x 1) h (x) = 2 + log x k (x) =-log x + 2 Properties of Common Logs 1. log 1 = 0 (because 10 0 = 1) 2. log 10 = 1 (because 10 1 = b) 3. log 10 x = x (because 10 x = 10 x ) 4. 10 logx = x (because log x = log x) From J. P. Wood, Spring 2012 6
Transformations of Logarithmic Functions (p.429) for f (x) = log b x Vertical: g (x) = log b x + c g (x) = log b x c Horizontal: g (x) = log b (x + c) g (x) = log b (x c) Reflection: g (x) = -log b x g (x) = log b (-x) Vertical Stretch/Shrink: g (x) = c log b x Horizontal Stretch/Shrink: g (x) = log b (c x) The Natural Logarithm: f (x) = log e x or f (x) = ln x Properties of Natural Logarithms 1. ln 1 = 0 2. ln e = 1 3. ln e x = x 4. e lnx = x Find the domain and x-intercepts of the following: a) f (x) = ln (x 2) b) g (x) = ln (2 x) From J. P. Wood, Spring 2012 7
4.3: Properties of Logarithms Properties of Logarithms Let a be positive not equal to 1, and let m and n be reals. Let u and v be positive reals and consider: log a (u) = m log a (v) = n The Product Rule: log a (uv) = log a (u) + log a (v) Ex, ln (uv) = ln (u) + ln (v) The Quotient Rule: u v log a ( ) = log a (u) log a (v) ln ( ) = ln (u) ln (v) Ex, u v Consider: log a u 2 = log a (u u)= log a u + log a u = 2 log a u The Power Rule: log a u n = n log a u ln u n = n ln u Ex: From J. P. Wood, Spring 2012 8
Use the Product Rule to expand the following expressions: Use the Quotient Rule to expand the following expressions: Use the Power Rule to expand the following expressions completely: Use log properties to expand the following expressions completely: From J. P. Wood, Spring 2012 9
Use log properties to condense the following expressions: Change-of-Base Formula Let a, b, and x be positive real numbers s.t. a 1 and b 1 Then log a x = log b x log b a Change base to Common Log: log a x = log x log a Change base to Natural Log: log a x = ln x ln a Graph f (x) = (½) x and g (x) = log ½ x log ½ x = ln x ln (½) now graph y = x From J. P. Wood, Spring 2012 10
4.4: Exponential and Logarithmic Equations Key Properties (1-1 properties): 1. M = N if and only if log b M = log b N 2. M = N if and only if b M = b N b > 0, b 1 Also, log b b x = x and ln e x = x b log b x = x and e ln x = x Solve the following exponential equations: 2 3x 8 = 16 27 x + 3 = 9 x 1 4 x = 15 10 x = 120000 e x = 7 e x + 5 = 60 40e 0.6x 3 = 237 5 x 2 = 4 2x + 3 4e 2x = 5 e 2x 3e x + 2 = 0 From J. P. Wood, Spring 2012 11
Solve the following logarithmic equations: ln x = 2 5 + 2 ln x = 4 2 ln 3x = 4 ln x ln (x 1) = 1 log 10 5x + log 10 (x 1) = 2 log 2 x + log 2 (x 7) = 3 ln(x + 2) ln(4x + 3) = ln( 1 x ) From J. P. Wood, Spring 2012 12
4.5: Exponential Growth and Decay; Modeling Data Common types of mathematical models involving exponential functions are: 1. Exponential growth: y = A 0 e kt, k > 0 2. Exponential decay: y = A 0 e -kt, k > 0 3. Logistics growth: y = a, b, c are constants with c > 0 and b > 0 2 c 1 + ae -bt 5. Logarithmic models: y = a + b ln x or y = a + b log x From J. P. Wood, Spring 2012 13
Modeling the growth of U.S. Population (in millions M): Year 1970 1980 1990 2000 2007 Population 203.3 226.5 248.7 281.4 300.9 Model a. Find an exponential growth function that models the data above. b. By which year will the U.S. population reach 315M? The exponential growth model is: y = A 0 e kt, with k > 0 Let t = 0 be the start of 1970. Then P 0 is 203.3M Thus, the exponential model that approximates this data is: P(t) = 203.3e kt, 0 < t < 37 In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 50 flies, and after 4 days there are 250 flies. How many flies will there be after 5 days? From J. P. Wood, Spring 2012 14
Carbon-14 Dating: a) Use the fact that after 5715 years, a given amount of carbon-14 will have decayed to half of the original amount to find the exponential decay model for carbon-14. b) Earthenware jars that contained the Dead Sea Scrolls had 76% of their original carbon-14. Estimate their age. The exponential growth model is: y = A 0 e kt, with k < 0 So, Ex 5, Spread of a Virus: In a town of 30,000 inhabitants, the function below models the number of people who have become ill from a flu virus after its initial outbreak: 30000 f (t) = 1 + 20e 1.5t, t > 0 a) How many people were ill when the epidemic began? b) How many people were ill after the fourth week c) What is the limiting size of the epidemic? From J. P. Wood, Spring 2012 15
Choosing a Model for Data: Weight-Loss Surgeries after 2001 x, number of Yrs after 2001 y, number of surgeries 1(2002) 2(2003) 3(2004) 4(2005) 5(2006) 63 103 141 171 178 Let s show the scatter plot on our TIs It looks logarithmic, so let s do an ln regression Select STAT, select CALC, select LnReg and Enter, next enter L 1, L 2, Y 1 and ENTER. (choose Y= from VARS, Y-VARS) y = 59.059 + 75.343 ln x now show graph over scatter plot From J. P. Wood, Spring 2012 16
Choosing another Model for Data: World Population in Billions in years after 1949 x, Year 1 11 21 31 41 51 57 y, Pop 2.6 3.0 3.7 4.5 5.3 6.1 6.5 Show scatter plot on TI Do a linear regression and an exponential regression f (x) = 0.073x + 2.316 and g(x) = 2.569(1.0170) x Which regression best models the data? Choosing y = ab x in base e y = ab x (ln b) x is equivalent to y = ae Rewrite the following exponential model in base e: g(x) = 2.569(1.017) x g(x) = Show on TI that function is identical From J. P. Wood, Spring 2012 17