Sec 5.1 Exponential & Logarithmic Functions (Exponential Models)

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2 Sec 5.1 Eponential & Logarithmic Functions (Eponential Models) 1. The population of the cit Suwanee, GA has consistentl grown b 4% for the last several ears. In the ear 000, the population was 9,500 people. Name: What would be the growth factor (multiplier)? If the trend continues what would be the population in 00?. Lisa purchases a house for $150,000 near Lake Jackson. The value of houses in the area where the house was purchased is averaging an increase of 6% per ear. What would be the growth factor (multiplier)? If the trend continues how much would the house be worth 1 ears after Lisa purchased the house? 3. Esther purchased a used car, a Ford Focus, for $8400. The car is epected to decrease in value b 0% per ear over the net couple of ears. What would be the deca factor (multiplier)? If the trend continues how much would the car be worth 6 ears after Esther purchased the car? 4. Freddie purchased a pair of never worn Vintage 1997 Nike Air Jordan XII Plaoff Black Varsit Shoe Sie 1 for $380. The shoes have shown an average growth rate of 14% per ear. What would be the growth factor (multiplier)? If the trend continues how much would the shoes be worth 5 ears after Freddie purchased the shoes? 5. A culture of bacteria triples b the end of each hour. There were initiall 50 bacteria present in the petri dish. What would be the growth factor (multiplier)? If the trend continues how man bacteria would there be 5 hours after the analsis began? M. Winking Unit 5-1 page 79

3 Number of Coins 6. Consider starting with pennies. Flip them both and for each one that lands heads up, add a penn to the pile. So, the pile should increase in sie. Again, flip the new pile of pennies which could be a sie of, 3, or 4. For ever penn that lands heads up add another penn to the pile. Repeat this process several times and record how the penn pile grows after each flip. Your values ma differ on Flips 3 and 4. Number of Flips Number of Pennies 0 Flip #1 Flip # Add a penn since this one landed heads up. Add a penn since this one landed heads up. Add another penn since this one landed heads up. Create a graph of the data. a. What is an appropriate growth factor (multiplier)? b. Create an equation that describes the relationship between the number of flips and the number of pennies in the pile. c. Approimatel how man pennies would there be on the 9 th flip? d. Should the graph be continuous or discrete? Eplain. e. What is an appropriate Domain and Range for the situation? Number of Coin Flips M. Winking Unit 5-1 page 80

4 7. Determine which of the following functions are eponential models of Growth and which are models of Deca. a. f() = (1.05) b. g() = 540 (0.9) + 1 c. h() = 4 ( 3 5 ) Circle the Answer Growth Deca Neither Circle the Answer Growth Deca Neither Circle the Answer Growth Deca Neither d. = 30 ( 7 5 ) e. = 400 e 5 f. = 9 () Circle the Answer Growth Deca Neither Circle the Answer Growth Deca Neither Circle the Answer Growth Deca Neither g. = 1 4 (3) h. p() = 50 e + 3 i. = 30 ( 5 ) Circle the Answer Growth Deca Neither Circle the Answer Growth Deca Neither Circle the Answer Growth Deca Neither 8. Consider the Compound Interest Formula: A = P (1 + r n )nt a. Determine the value of an account in which a person invested $6000 for 1 ears at an annual rate of 9% compounded annuall (n = 1). A = Value of Account after Compounding P = Original Amount Invested r = Annual Interest Rate as a decimal n = Compounds per Year t = Number of Years Interest is Accrued n = 1 :Annuall n = :Semi-Annuall n = 4 :Quarterl n = 1 :Monthl n = 5 :Weekl n = 365 :Dail b. Determine the value of an account in which a person invested $6000 for 1 ears at an annual rate of 9% compounded quarterl (n = 4). c. Determine the value of an account in which a person invested $6000 for 1 ears at an annual rate of 9% compounded weekl (n = 5). 9. Consider the Compound Interest Formula: A = P e rt Determine the value of an account in which a person invested $6000 for 1 ears at an annual rate of 9% compounded continuousl. A = Value of Account after Compounding P = Original Amount Invested r = Annual Interest Rate as a decimal t = Number of Years Interest is Accrued

5 Sec 5. Eponential & Logarithmic Functions (Graphing Eponential Functions) 1. Consider the eponential function, f() = 3. A. Fill in the missing values in the table below. B. Plot the points from the table and sketch a graph Label an asmptotes. 0 f() Name: C. Determine the Domain & Range of the function.. Consider the eponential function, g() =. A. Fill in the missing values in the table below. B. Plot the points from the table and sketch a graph Label an asmptotes. 3 g() C. Determine the Domain & Range of the function. 3. Consider the eponential function, h() = ( 1 ) + 1. A. Fill in the missing values in the table below. B. Plot the points from the table and sketch a graph Label an asmptotes. 3 1 h() 0 1 C. Determine the Domain & Range of the function. 3 M. Winking Unit 5- page 8

6 4. Determine the asmptote and sketch a graph (label the an intercepts, points when = 0, and when = 1.) A. f() = 3 4 B. g() = ( 1 ) + C. h() = 3 5. Create two different eponential functions of the form f() = a b + c that have a horiontal asmptote at = Given the function f() is of the form f() = a b + c, has a horiontal asmptote at = 1, and passes through the point (0,), create a possible function for f(). 7. Consider t() is of the form t() = a + c. 8. Consider w() is of the form w() = a + c. Which of the following must be true for the parameter a? a > 1 0 < a< 1 a <0 Which of the following must be true for the parameter c? Which of the following must be true for the parameter a? a > 1 0 < a < 1 a <0 Which of the following must be true for the parameter c? c > 0 c = 0 c <0 M. Winking Unit 5- page 83 c > 0 c = 0 c <0

7 9. Determine the -intercept and -intercept of the following eponential functions: a. r() = 3 6 b. r() = The parent graph is shown in light gra on the graph. Graph the transformed function on the same Cartesian coordinate grid and describe the transformations based on the function t(). a. Parent Function: f() = b. Parent Function: f() = Transformed Function: t() = ( ) 6 Transformed Function: t() = ( 4) Determine the Domain & Range of the function. Determine the Domain & Range of the function. c. Parent Function: f() = 3 d. Parent Function: f() = 3 Transformed Function: t() = 3 (+3) Transformed Function: t() = 3 ( ) + Determine the Domain & Range of the function. Determine the Domain & Range of the function. M. Winking Unit 5- page 84

8 11. Given the graph of f() on the left, determine an equation for g() on the right in terms of f(). a. g() = b. g() = c. g() = M. Winking Unit 5- page 85

9 1. Given a table of values for the eponential function f() and a description of the transformations for the function g(), fill out the table of values based on the original points for g(), the transformed function. a f() ⅓ Translated Down g() b f() ½ Translated Left 1 & Up g() c f() ¼ Reflect over -ais g() d f() ½ Vertical Stretch of Factor 3 g() 13. Given each of the graphs below are eponential functions of the form f() = a, determine the parameter a in each graph. a. b. f() = h() = M. Winking Unit 5- page 86

10 Sec 5.3 Eponential & Logarithmic Functions (Converting Between Eponents & Logs) Name: 1. Rewrite the following eponential statements as logarithmic statements. (EXP LOG) a. 15 = 5 3 b. 6 = 64 c. 4 = 16 d. 43 = 5 e. e = 9 f. = e 5. Rewrite the following logarithmic statements as eponential statements. (LOG EXP) a. 3 = log (8) b. 5 = log (43) c. log 6 () = 3 d. ln() = 5 e. log 4 (56) = f. = ln(3) M. Winking Unit 5-3 page 87

11 3. Evaluate the following basic logarithm statements. a. log (3) b. log 7 (49) c. log 6 (6) d. log 4 (56) e. log(1000) f. ln(e 7 ) 4. Evaluate the following logarithm statements. a. log 5 (5 1 ) b. (log 3 (3 )) c. log 3 (9 3 ) d. log (16 5 ) e. 4 log 4 (16) f. 3 log 3 (81) d. 5 log 5 (1) e. 4 log (3) f. e ln(5) M. Winking Unit 5-3 page 88

12 Evaluate the following using the prime factoriation of 9 4. Evaluate the following using a recognied propert. log 3 (9 4 ) log 3 (9 4 ) 5. Rewrite each of the following using the propert above. a. log 5 (5 3 ) b. log 3 (14 5 ) c. ln(9 3 ) Evaluate the following with our calculator b changing the base to 3 decimal places (show the work to provide reasoning) log 9 = 6. Evaluate the following with our calculator b changing the base to 3 decimal places a. log 5 (50) b. log 8 (1) c. log 4 ( ) d. log 3 (1) e. log(53) f. ln(8) M. Winking Unit 5-3 page 89

13 Sec 5.4 Eponential & Logarithmic Functions (Graphing Logarithmic Functions) 1. Consider the logarithmic function, f() = log (). A. Fill in the missing values in the table below. B. Plot the points from the table and sketch a graph Label an asmptotes. 0 f() Name: ½ 1 C. Determine the Domain & Range of the function. 4 ¼ D. Determine the End Behavior.. Consider the logarithmic function, g() = log ( + 3) + A. Fill in the missing values in the table below. B. Plot the points from the table and sketch a graph Label an asmptotes. 3 g().5 1 C. Determine the Domain & Range of the function. 0 1 D. Determine the End Behavior Consider the logarithmic function, h() = ln( 1) + 3. A. Fill in the missing values in the table below. B. Plot the points from the table and sketch a graph Label an asmptotes. h() 1 1. C. Determine the Domain & Range of the function D. Determine the End Behavior. 5 M. Winking Unit 5-4 page 90

14 4. Determine the asmptote and sketch a graph (label the an intercepts, points when ou locate log(1)). A. f() = log ( + 3) B. g() = log 5 ( ) 1 C. h() = ln( + 1) 5. Create two different logarithmic functions of the form f() = a log ( + b) + c that have a vertical asmptote at = Given the function f() is of the form f() = log ( + b) + c, has a vertical asmptote at = 1, and passes through the point (0,), create a possible function for f(). 7. Consider t() is of the form t() = a log ( + b). 8. Consider w() is of the form w() = a log ( + b) Which of the following must be true for the parameter b? b < 1 b = 0 b > 0 Which of the following must be true for the parameter a? Which of the following must be true for the parameter b? b < 0 b = 0 b > 0 Which of the following must be true for the parameter a? a < 0 a = 0 a > 0 M. Winking Unit 5-4 page 91 a < 0 a= 0 a >0

15 9. Determine the -intercept of the following logarithmic functions: a. r() = log 3 ( + 9) b. p() = log 3 ( ) c. m() = log 5 ( + 1) + 9 Consider the parent function of f() = log m (). The following would be a transformed function t() = a log m (b( c)) + d a > 1: Vertical Stretch (eg. a = 3) (factor a ) 0 < a < 1:Vertical Compress (e.g. a = 0.) (factor a ) -1 < a < 0: Reflect over -ais & Vertical Compress (e.g. a =- 0.) (factor a ) a = -1: Reflect over -ais a < -1: Reflect over -ais & Vertical Stretch (e.g. a =- 4) (factor a ) b > 1: Horiontal Compress (eg. b = 3) 0 < b < 1: Horiontal Stretch (e.g. b = 0.) -1 < b < 0: Reflect over -ais & Horiontal Stretch (e.g. b =- 0.) b = -1: Reflect over -ais b < -1: Reflect over -ais & Horiontal Compress (e.g. b =- 4) c = Horiontal d = Vertical Translation Translation (opposite direction) 10. Describe the transformations based on the function t(). a. Parent Function: f() = log 3 () b. Parent Function: f() = ln() Transformed Function: t() = 3 log 3 ( + ) 1 Transformed Function: t() = ln(( + 4)) 11. Given a table of values for the eponential function f() and a description of the transformations for the function g(), fill out the table of values based on the original points for g(), the transformed function. a f() Undefined Translated Down g() b f() Undefined Translated Left 1 & Up g() M. Winking Unit 5-4 page 9

16 1. The parent graph is shown in light gra on the graph. Graph the transformed function on the same Cartesian coordinate grid and describe the transformations based on the function t(). a. Parent Function: f() = log () b. Parent Function: f() = log () Transformed Function: t() = log ( ) + 3 Transformed Function: t() = 3 log ( + 4) Determine the Domain & Range of the transformed function. Determine the Domain & Range of the transformed function. 13. Given the graph of f() on the left, determine an equation for g() on the right in terms of f(). a. g() = 14. The graph below is a functions of the form f() = log a, determine the parameter a. 15. The graph below is a functions of the form g() = log a ( + b), determine the parameter b. f() = g() = M. Winking Unit 5-4 page 93

17 Sec 5.5 Eponential & Logarithmic Functions (Inverses of Eponential and Log Functions) Name: 1. Consider the eponential function f() shown below. Find the inverse of the function, sketch a graph of the inverse, and determine whether or not the inverse is a function. A. Graph of Inverse Is the Inverse a Function? YES NO B. Graph of Inverse Is the Inverse a Function? YES NO C. Graph of Inverse Is the Inverse a Function? YES NO D. Graph of Inverse Is the Inverse a Function? YES NO M. Winking Unit 5-5 page 94

18 . Consider the logarithmic function f() shown below. Find the inverse of the function, sketch a graph of the inverse, and determine whether or not the inverse is a function. A. Graph of Inverse Is the Inverse a Function? YES NO B. Graph of Inverse Is the Inverse a Function? YES NO C. Graph of Inverse Is the Inverse a Function? YES NO D. Graph of Inverse Is the Inverse a Function? YES NO M. Winking Unit 5-5 page 95

19 Sec 5.6 Eponential & Logarithmic Functions (Properties of Eponents and Logarithms) Name: 3 = 3 = = Simplif a. (5 )(14 3 ) b c. 3 3 d e. ) n (4m ) (-3m n f c b 16 a c b 4 a = 5 ( ) ( ) = 6 M. Winking Unit 5-6 page 96

20 (1 continued) Simplif g a b c 4 a b h i. 5a 3 b c j. k. l. (9a 3 b 5 )( 4a 3 b 7 ) m m n p 6m n n. M. Winking Unit 5-6 page 97

21 Rules of Logarithms a. log 8 + log 4 b. log 3 81 log 3 3 c. log (8 3 ) d. log c a + log c b e. log d log d f. b log t (a). Rewrite the following as a single logarithm epression and simplif. a. log (40) log (10) b. log 5 (30) + log 5 () log 5 (4) c. ln(8) ln() d. log 3 (4) + log 3 ( ) log 3 () e. log b (3) + 3 log b () log b ( ) f. ln() + 3 ln( 3 ) ln(6) M. Winking Unit 5-6 page 98

22 3. Epand each of the single logarithm epressions in to multiple logarithms. a. log 5 (1 3 ) b. ln ( 43 3 ) c. log ( 8a3 b ) d. ln (3a ) c b M. Winking Unit 5-6 page 99

23 Sec 5.7 Eponential & Logarithmic Functions (Solving Eponential Equations) Name: 1. Solve the following basic eponential equations b rewriting each side using the same base. a. 3 1 = 81 b. 3 = 18 c. 1 = 4 3. Solve the following basic eponential equation b rewriting each as logarithmic equation and approimating the value of. a. 4 = 10 b. 5 3 = c. e = Solve the following eponential equation b rewriting each as logarithmic equation and approimating the value of. a. 6 8 = 11 b = c. 3e + = 9 d = 8 M. Winking Unit 5-7 page 100a

24 4. Solve the following eponential inequalities. a. 3 7 > 0 b c. e + 1 < 33 d Solve the following applications a. Create an equation that represents the value P of an investment t ears after the initial investment. The initial investment was $300 and increases b1% each ear (compounded annuall). This would suggest that the account value could be modeled b P = 300(1.1) t. Determine how man ears it should take for the investment to double in value. b. There are 15 virus particles known as a virion in a host and the number of virions doubles ever hour and continues this model for the first 8 hours. The number of N virion after t hours can be found using the formula N = 15 ( t ). How long will it take approimatel for there to be 4,500,000 virions living in the host? c. There are initiall 8 frogs living in a pond in the back of a farm. The number frogs can described b the formula P = 8 e (0.t). How man ears will it take for the population of frogs to grow to 100 if the model continues? M. Winking Unit 5-7 page 100b

25 Sec 5.8 Eponential & Logarithmic Functions (Solving Logarithmic Equations) 1. Solve the following basic logarithmic equations. a. log (3 + 3) = log (5 15) b. log 5 ( 1) = log 5 (4) Name:. Solve the following basic eponential equation b rewriting each as logarithmic equation and approimating the value of. a. log () = 7 b. log 3 (4 + 1) = 4 c. ln( 3) = 5 3. Solve the following eponential equation b rewriting each as logarithmic equation and approimating the value of. a. log (3 + ) + 5 = 4 b. log 6 (9) + log 6 (4) = 6 c. 3 ln( + 1) = d. ln(1) ln() = 8 M. Winking Unit 5-7 page 101a

26 4. Solve the following eponential inequalities. a. log (3 ) > 4 b. log 3 (9 + 9) 4 c. ln( + ) < 1 d. log 4 (4 + 0) 3 5. Solve the following applications a. The population of trout in a lake could be modeled b the equation P = 50 log (t + ) where P is the number of fish and t is the number of ears after 016. If the trend continues, how man ears after 016 will it take for the population to reach 600 trout? b. The Richter scale measures the magnitude of an earthquake based on the amount energ determined b the ground motion from a set distance from the epicenter of the quake. The Magnitude is given b = E log ( ). If the Magnitude of an earthquake was 7., how much energ was released b the earthquake? M. Winking Unit 5-7 page 101b