PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1: Exponential and Logistic Functions

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1 PRE-CALCULUS: Finne,Demana,Watts and Kenned Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.1: Eponential and Logistic Functions Which of the following are eponential functions? For those that are eponential functions, state the initial value and the ase. For those that are not, eplain. Eponential Function A function that can e rewritten in the form a, where a is non-zero, is positive, and 1. a: initial value at = 0 : ase A) f() = 3 B) g() = 6 - C) h() = 21.5 D) h( ) 72 E) f ( ) 56 Compute the eact value of the function without using a calculator A) 2 when = 0 B) 2 when = -3 C) 2 when = 1/2 D) 3 8 when = -2/3 1 P a g e

2 Determine a formula for the eponential function g() and h() whose values are given in the tale g() h() -2 / / /2 Given 2 points on the graph of an eponential function, find the formula A) (0, 2) (2, 18) B ) 0.3 3, 3 e 2 P a g e

3 3 P a g e Sketch a graph of the following functions in the same viewing window [-2, 2] [-1, 6] ) Determine the domain and range 1) Determine the domain and range 2) Is the function even, odd or neither 2) Is the function even, odd or neither 3) Intervals of Increase or Decrease 3) Intervals of Increase or Decrease ) Find an etrema. ) Find an etrema. 5) Determine the end ehavior 5) Determine the end ehavior 6) Find an asmptotes 6) Find an asmptotes 7) Intervals of Concavit 7) Intervals of Concavit

4 Descrie how to transform the graph of f() = 2 into the graph of g a) g() = 2-1 ) g() = 2 - c) g( ) 2 3 d) g() = 2 3- Descrie how to transform the graph of f() = e into the graph of g a) g() = e ) g() = e - c) g ( ) 3e 1 d) g() = e 2-2 P a g e

5 State whether the function is eponential growth or deca and descrie its end ehavior A) f 3 ( ) 2 B) f ( ) 3 2 C) f 1 ) ( D) 1 f ( ) Graph the following functions on our calculator. Find the -intercept and the horizontal asmptotes f ( ) e f ( ) 3 5 P a g e

6 Sketch a graph of the following functions e 1) Determine the domain and range 1) Determine the domain and range 2) Is the function even, odd or undefined 2) Is the function even, odd or undefined for < 0 for < 0 3) Intervals of Increase or Decrease 3) Intervals of Increase or Decrease ) Find an etrema. ) Find an etrema. 5) Determine the end ehavior 5) Determine the end ehavior 6) Find an asmptotes 6) Find an asmptotes 7) Intervals of Concavit 7) Intervals of Concavit 6 P a g e

7 Sketch a graph of the following functions 1 2e 1 2e 1) Determine the minimum and 1) Determine the minimum and Maimum capacit (Horizontal As) Maimum capacit (Horizontal As) 2) Determine the -intercept 2) Determine the -intercept 3) Determine the domain and range 3) Determine the domain and range ) Intervals of Increase or Decrease ) Intervals of Increase or Decrease 5) Determine the end ehavior 5) Determine the end ehavior 6) Find an asmptotes 6) Find an asmptotes 7) Determine Half the ma capacit 7) Determine Half the ma capacit 8) Intervals of Concavit 8) Intervals of Concavit 7 P a g e

8 Using the data in the tale elow and assuming the growth is eponential, answer the following questions? Cit 2000 Population 2007 Population Austin, TX 65, ,562 San Jose, CA 898, ,899 a) When will the population of San Jose, California, surpass 1 million persons? ) In what ear will the population of Austin, TX and San Jose, CA e the same? 8 P a g e

9 Based on recent Census Data, a logistic model for the population of Dallas, TX, t ears after 1900, is as follows: e a) What was the population of Dallas, TX in the ear 2000? ) According to the model, what is Dallas maimum sustainale population? c) According to this model, when was the population 1 million. 9 P a g e

10 Bacteria Growth The numer of acteria after t hours is given 150 e 0.521t a) What was the initial amount of acteria present? ) How man acteria are present after hours? c) How man hours will it take until there are 00 acteria? 10 P a g e

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12 Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.2: Eponential and Logistic Modeling Tell whether the function is an eponential growth or deca function. Then find the constant percentage rate of growth or deca. 10. t t t t Determine the eponential function that satisfies the given conditions A) Initial Value 12, increasing at the rate of 8% per ear B) Initial Value 12, decreasing at the rate of 8% per ear C) Initial Population 200, tripling ever 5 das D) Initial Mass 20 grams, cutting in fourths once ever 3 das 12 P a g e

13 Determine a formula for the eponential function whose values are given 20) g() Determine a formula for the eponential function whose points are given 21) (0, ) (5, 8.05) Find the logistic function that satisfies the given conditions A) Initial Value 6: Ma Capacit (Limit to growth) = 30 Passing through (1, 15) 13 P a g e

14 B) Initial Population = 20, Ma Capacit (Limit to growth) = 100 Passing through (, 75) 32. Eponential Growth: The population of River Cit in the ear 1910 was 200. Assume the population increased at a rate of 2.25% per ear. a) Estimate the population in ) Predict when the population reached 20,000. Eample : Suppose the half-life of a certain radioactive sustance is 20 das and there are 5 grams present initiall. A) Epress the amount of the sustance remaining as function of time. B) Find the time when there will e 1 gram of the sustance remaining. 1 P a g e

15 Watauga High School has 1200 students. Bo, Carol, Ted and Alice start a rumor, which spreads logisticall so that 1200 S( t) 0. 9t models the numer of students who have heard the 1 39e rumor the end of da t. A) How man students have heard the rumor the end of Da 0. B) How long does it take for 1000 students to hear the rumor? Use the data in the tale and eponential regression to predict Dallas, TX population in , , , , ,006, ,1888, P a g e

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17 PRE-CALCULUS: Finne,Demana,Watts and Kenned Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.3: Logarithmic Functions and their graphs What ou'll Learn Aout Changing etween Logarithmic and eponential form: Find the inverse function for = 2 If 1, then log 0, if 0 and and onl if If log log 0, Properties: 0 1, and an real numer 0 log 1 0 ecause 1 log 1ecause ecause ecause log 1 log Evaluate the logarithmic epression without using a calculator a) log 8 c) log ) log 3 3 d) log 1 e) log P a g e

18 Evaluate the logarithmic epression without using a calculator a) log100 ) log 5 10 c) 1 log 100 d) ln e e) ln e 5 f ) ln 5 e Evaluate the logarithmic epression without using a calculator a) 6 log 6 11 ) 10 log6 c) e ln 18 P a g e

19 Use a calculator to evaluate the logarithmic epression if it is defined and check our result evaluating the corresponding eponential epression a) log 3.5 = ) log 0.3 = c) log (-3) = d) ln 23.5 = e) ln 0.8 = f) ln(-5) = Solve the equation a) log = 3 ) log P a g e

20 Descrie how to transform the graph of = ln into the graph of the given function. Sketch the graph hand. a) g ( ) ln( 2) 1 ) h( ) ln(3 ) 1) Determine the vertical asmptotes 1) Determine the vertical asmptotes 2) Determine the -intercept 2) Determine the -intercept 3) Determine the domain and range 3) Determine the domain and range ) Intervals of Increase or Decrease ) Intervals of Increase or Decrease 5) Determine the end ehavior 5) Determine the end ehavior 6) Intervals of Concavit 6) Intervals of Concavit 20 P a g e

21 Descrie how to transform the graph of = ln into the graph of the given function. Sketch the graph hand. a) g( ) 3log ) h ( ) log( ) 2 1) Determine the vertical asmptotes 1) Determine the vertical asmptotes 2) Determine the -intercept 2) Determine the -intercept 3) Determine the domain and range 3) Determine the domain and range ) Intervals of Increase or Decrease ) Intervals of Increase or Decrease 5) Determine the end ehavior 5) Determine the end ehavior 6) Intervals of Concavit 6) Intervals of Concavit 21 P a g e

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23 Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.: Properties of Logarithmic Functions 3.1: Eponential What ou'll and Learn Logistic Aout Functions Use our Calculator to Determine which of the following are True. 1. log (5 + 2) = log 5 + log 2 2. log( 5 2) log5 log log (5-2) = log 5 - log 2. log log5 log log( 5 2) 2log log 2 log5 log 2 7. log( 5 2 ) log5 log 5 8. log( 5 2 ) 2log 5 9. ln ( + 2) = ln + ln log( 7) 7log 11. log( 5) log 5 log 12. ln ln ln log log log 3 1. log 3log 15. ln( 2 ) ln ln 16. log log log 23 P a g e

24 Let,R, and S are positive real numers with 1, and c an real numer log ( RS ) log log R log log S R c c log R log R log R S S Prove the Product Rule for Logarithms: Let R and log S log log ( RS) log R log Assuming and are positive, use properties of logarithms to write the epression as a sum or difference of logarithms or multiples of logarithms 5 A ) log(8 ) B) ln S C ) log 2 5 D ) log(8 2 ) E) ln P a g e

25 Let,R, and S are positive real numers with 1, and c an real numer log ( RS ) log log R log log S R c c log R log R log R S S Assuming, and z are positive, use properties of logarithms to write the epression as a single logarithm A ) log log6 B ) ln ln C) log D) 6log log 2 E) 5log 2 2 3log z F) ln 5 2ln 25 P a g e

26 26 P a g e 2 6 log 096 log 096 log ecause log 096 log 096 log 6 ecause

27 For positive real numers a, and with a 1and 1 log log log a a ln ln Rewrite the following as an eponential function then solve for A ) log 7 Use the change of ase formula and our calculator to evaluate the logarithm A log 16 B ) log ) 3 Epress using onl natural logarithms ) g() log B) g() log A P a g e

28 Descrie the transformation of each function from the original function ln() or log() 2) f() = ln() + 2 ) f() = ln(-) + 2 6) f() = ln(5 ) A) f() = ln( 5) 52) f() = -3log(1-) + 1 B) = log C) = log(2) D) log 28 P a g e

29 Sketch a graph of the following functions log f ( ) ln f ) 5 ( 3 1) Determine the vertical asmptotes 1) Determine the vertical asmptotes 2) Determine the -intercept 2) Determine the -intercept 3) Determine the domain and range 3) Determine the domain and range ) Intervals of Increase or Decrease ) Intervals of Increase or Decrease 5) Determine the end ehavior 5) Determine the end ehavior 6) Intervals of Concavit 6) Intervals of Concavit 29 P a g e

30 Sketch a graph of the following functions f ( ) log 3 f ( ) ln 1) Determine the vertical asmptotes 1) Determine the vertical asmptotes 2) Determine the -intercept 2) Determine the -intercept 3) Determine the domain and range 3) Determine the domain and range ) Intervals of Increase or Decrease ) Intervals of Increase or Decrease 5) Determine the end ehavior 5) Determine the end ehavior 6) Intervals of Concavit 6) Intervals of Concavit 30 P a g e

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32 Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.5: Equation Solving and Modeling What ou'll Learn Aout Find the eact solution algeraicall, and check it sustituting into the original equation A ) B ) ) 23 2 C 6 D ) E ) log 5 F ) log P a g e

33 Solve each equation algeraicall 0.03 A ) B ) 50e 500 C ) 2ln D ) 2 log P a g e

34 Solve each equation. A ) log 3 9 B ) C ) 3e 2 e 0 3 P a g e

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36 1 E ) ln( 2) ln( ) 0 2 F ) log( 2) log( 3) log2 36 P a g e

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