Day Date Assignment. 7.1 Notes Exponential Growth and Decay HW: 7.1 Practice Packet Tuesday Wednesday Thursday Friday

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1 1 Day Date Assignment Friday Monday /09/18 (A) /1/18 (B) 7.1 Notes Exponential Growth and Decay HW: 7.1 Practice Packet Tuesday Wednesday Thursday Friday Tuesday Wednesday Thursday Friday Monday /1/18 (A) /14/18 (B) /15/18 (A) /16/18 (B) /0/18 (A) /1/18 (B) //18 (A) 0//18 (B) 0/6/18 (A) 7. Notes Functions involving e HW: 7. Practice Packet 7. Notes Evaluating logs and ln HW: 7. Practice Packet 7.4 Notes Inverse and Graphing Logs HW: 7.4 Practice Packet ACT Review ACT Review Tuesday /7/18 ACT Testing for juniors-no school for everyone else Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday 7.1 Practice Problems /8/18 (B) /01/18 (A) /0/18 (B) /05/18 (A) /06/18 (B) /07/18 (A) /08/18 (B) /09/18 (A) /1/18 (B) /1/18 (A) /14/18 (B) /15/18 (A) /16/18 (B) /19/18 (A) /0/18 (B) /1/18 (A) //18 (B) 7.4 Notes Inverse and Graphing Logs HW: 7.4 Practice Packet Review Day HW: Practice Test Part 1 Unit 7 Test Part Notes Applying Properties of Logs HW: 7.5 Practice Packet 7.6 Notes Solving Exponential Equations HW: 7.6 Practice Packet 7.7 Notes- Solving Log Equations HW: 7.7 Practice Packet 7.8 Notes- Modeling with Logs HW: 7.8 Practice Packet Review Day HW: Practice Test part Unit 7 Test Part Graph the following exponential functions and describe their transformations from the graph g(x) =4 x. Also state the domain range, and end behavior. 1. f(x) = 4 x +. f(x) = -4 x f(x) = 4 x You deposit $500 in an account that earns.5% annual interest. Find the balance after three years if the interest is compounded with the given frequency. 4. annually 5. Quarterly 6. Monthly 7. Daily

2 8. From 1990 to 000, the population of California can be modeled by P = 9,816,591(1.018) t where t is the number of years since a) What was the population in 1990? b) What is the growth factor and annual percent increase? c) Estimate the population in 007. Tell whether the following functions represent exponential growth or exponential decay. 9. f(x) = 5( 4 5 )x 10. f(x) = 5() -x 11. f(x) = 7 x Graph the following exponential functions and describe their transformations from the graph (x) = ( 1 )x. Also state their domain, range, and end behavior. 1. f(x) = -( 1 )x+1 1. f(x) = ( 1 )x Describe the domain and range of the function f(x) = 1 x 9 + A. Domain: (, 9) B. Domain: (0, 9) C. Domain: (, + ) D. Domain: (, + ) Range: (, + ) Range: (, 0) Range: (, 9) Range: (, + ) 15. Which of the following expressions simplifies to y y? A. y 7 B. (4x y 1 4 y 5 ) (x 4 y 1 ) 4 C. y1 y D. 9x7 y y 1 xy 16. When evaluating the function f(x) = 5 x+ 6 for any real number x, what must be true about the value of f(x)? A. The value of f(x) is always greater than B. The value of f(x) is always greater than 6 C. The value of f(x) is always negative D. The value of f(x) is always positive Factor each expression completely x 45x 18. x 4 1x x + x 0. Solve: 9x + 5x 15 = x 1. Find f(f(x)) if f(x) = 7x 1.. A neighborhood recreation program serves a total of 80 children who are either 11 years old or 1 years old. The sum of the children s ages is,8 years. How many 11-year-old children does the recreation program serve? A. 55 B. 1 C. 1 D. 158 E Practice Problems Simplify the following expressions using the natural base e (a) without a calculator, and (b) with a calculator. 1. (e ). e -5 e -. ( e 6e ) Simplify the following expressions using the natural base e. 4. e x 6e x 1 e x 5. 5e x+6 e x 1 e 5x

3 Graph the following functions and describe their transformations from the graph y = e x. Also state the domain and range. 6. f(x) = ½ e x 7. f(x) = - e x You deposit $500 in an account that pays 5.5% annual interest compounded continuously. Find the balance after each amount of time. 8. years years 10. Find the amount of interest earned in # 8 & 9 Simplify each expression (no decimal answers). Write all answers with positive exponents. Assume all variables have positive values. Show your work! z 5 Factor completely. z 54z x x x x 6x x + 1x A particular jeweler uses the formula d = 4w to relate the average diameter (d) of a cultured pearl in millimeters to its weight (w) in carats. The jeweler sells the pearls to customers for $.5 per carat. How much would a cultured pearl with a 9.5 mm average diameter cost? 0. Solve for x: x = 5 x + 1. Solve for x: x 5 = x + 7. In the figure below, A, B, C, and D are collinear, FC is parallel to ED, BE is perpendicular to ED, and the measures of FAB and EBA are as marked. What is the measure of FCB? A. B. 57 C. 6 D. 84 E. Cannot be determined from the given information 7. Practice Problems Rewrite the following in either exponential form or logarithmic form 1. log 81 = = 1. log 50. = ( 1 5 ) 1 = 5 Evaluate the logarithmic functions without a calculator. 5. log 4 6. log log 44 / 8) log ) ln e Evaluate the functions with a calculator. 10. log log 1 1. ln.5 Simplify the following using inverse properties 1. 7 log 7 x 14. log 1111 x 15. log 66 x ln 4x 16. e 17. log 5 + ln(e 1 ) log 4 64 log e ln log log1 8

4 4 19. How much money must be deposited now in an account paying 5.% annual interest, compounded weekly, to have a balance of $1500 after 8 years? Simplify completely without using a calculator. 0) ln e -17 1) log 16 5x ) log ln 9x ) e 4) Charles has $15,000 that he wants to invest for 4 years. He can either invest the money with Bank #1, earning.4% compounded monthly, or he can invest the money with Bank #, earning.% compounded continuously. Which bank should he use? Show your calculations and justify your conclusion. 5. Which of the following equations represent exponential decay? (you may choose more than one) I. f(x) = ex III. f(x) = 5e 7x II. f(x) = 0.5e 4x IV. f(x) = 10e x 6. For all nonzero real numbers p, t, x, and y such that, which of the following expressions is equivalent to t? F. G. H. I. J. 7.4 Practice Problems Find the inverse of the following functions 1. y = log 7 x. y = ln (x 1). F(x) = log 8 x 1 4. y = ln x + Graph the following functions, and state their transformations (for #5, 7 from y = log 8 x; for 6, 8 from y = ln x) and their domain and range. 5. y = log 8 (x + 1) 6. y = - ln (x + 1) 7. f(x) = log 8 (x ) y = ½ ln x Describe the transformation from the parent function, y = x, identify the domain and range, and then sketch the graph. 9. y = x y = - x 1 + Describe the transformation from the parent function, y = x, identify the domain and range, and then sketch the graph. 11. y = x y = x + 5 Rewrite each equation as a logarithmic or exponential equation. Then solve for x. 1) log 5 x= 14) log 7 = x 15) ln x = 0 16) 4 x = Which of the following sets of equations are NOT inverses of each other? A. y = log (x 1) y = x + 1 C. y = x + 4 y = log x 4 B. y = log x + 7 y = x 7 D. y = 5 x+ y = log 5 x 18. Simplify : log 81 ln(e 7 ) log log 5 65

5 5 19. The graph of which function is vertically stretched by a factor of 6 and translated units right from the graph of the parent function? A. y = 6 log (x ) C. y = log (6x + ) B. y = 6( x+ ) D. y = 6x Graph the following functions, and describe the transformations (for #0 1, from h(x) = log x; for #, from g(x) = ln x) and the domain and range. 0. y = log (x + 1) 1. f(x) = log (x ) +. f(x) = ln (x ). f(x) = - ln (x 1) + 4. If n = 8 and 16 m = 4 n 8, then m =? F. -4 G. - H. 0 I. 1 J. 8 Practice Test Part 1 Graph the following function and describe the transformations from their parent function. Then state the domain and range. 1. parent function: y = 4 x Graph: y = 4 (x ) + 5. Parent function: y =( 5 )x Graph: y = - ( 5 )(x+1) -. Parent function: y = e x Graph: y = e x Parent function: y = log x Graph: f(x) = log (x + 1) 6 5. Parent function: y = ln x Graph: f(x) = -1/ ln x 6. Find the value of $1000 deposited for 10 years in an account paying 7% annual interest compounded daily. 7. Find the value of $1000 deposited for 8 years in an account paying 8% annual interest compounded weekly. 8. Write the equation log 4 79 = 6 in exponential form Write the equation x = 100 in logarithm form. Evaluate the following without using a calculator: 10. log log log 4 1. ln e 14. ln5 e 15. log ln e log log log 10 eln log 4 56 Find the inverse of each function: 18. y = ln (x + ) 19. y = 4 x 0. y = log 6 (x 1) 1. Is y =.9 x an example of exponential growth or decay? How about y = -x? 7.5 Practice Problems Evaluate the following logarithmic expressions using log and log log 8. log log log 11

6 6 Expand the following expressions 5. log x 6. log 5 6x 4 Condense the following expressions y 7. ln xy 8. ln 1 x 5x 9. log 6 y 10. log + ½ log x log ln x ln + ln 6 1. ln(x + 1) ln y + ln y + ln 1. log 4 + log x + log y 14. log 9 + log log Evaluate the following using the change of base formula (give exact and approximate solutions): 15. log log log. 18. log 6 4 You deposit $500 in an account that pays 5.% annual interest. Find the balance after 4 years if the interest is compounded: 19. monthly 0. Daily 1. continuously Solve each equation. Simplify radical answers. Show your work and check for extraneous solutions.. x 7.6 Practice Problems 9 = -1. x 6 = x 8 4. (x 4) / 9 = 16 Solve the following exponential and logarithmic equations. For #1 4, give both decimal and exact answers. 1. e x = 4. 9 x = x + 1 = 4. ( x + 6 ) + = x = x 1 = 1000 x ( 5 )x+7 = ( x = 4 x x = 17 + x 5 ) x = ( 7 64 )x How long would it take for an investment of $9,000 to grow to $1,000, if it is in an account that earns 6% interest, compounded continuously? 1. How long would it take for $00 to earn $50 in interest in an account earning 5.5% interest, compounded annually? 1. How much money must be deposited now in an account paying 7% annual interest, compounded monthly, to have a balance of $000 after 5 years? Solve each equation. If needed, round to decimal places x 7 = e 0.07t = e x = x + = x Solve for x:.5 8x 4 = ( 15 8 )x Solve: 7 x+1 = 6 x+5 0. Scientists experimenting with the effects of a new antibiotic on a particular bacteria population found that the a population of bacteria can be modeled with the function f(x) = 000(1 0.5) t, where t is the time in days the antibiotic is taken. Scientists have also discovered that this antibiotic can only be taken for a maximum 5 days before it is considered harmful to the patient. In order to consider a person cured of the bacterial infection, an initial population of 000 bacteria must be reduced to less than 00. Is it possible to cure a person with the new antibiotic?

7 7 1. A sample of two bacteria strains are being studied at a lab. After h hours, the population of Bacteria M is modeled by M(h) = 0(1.8) h, and Bacteria N is modeled by N(h) = 0(1.65) h. When is the population of Bacteria M greater than the population of Bacteria N? A. The population of Bacteria M is always greater than the population of Bacteria N. B. The population of Bacteria M is never greater than the population of Bacteria N. C. The population of Bacteria M is greater until a point between 4 and 5 hours, after which Bacteria N has the greater population. D. The population of Bacteria N is greater until a point between 4 and 5 hours, after which Bacteria M has the greater population. 7.7 Practice Problems Solve the following logarithmic equations. Check for extraneous solutions. 1. log 7( x) = log 7 5x. 7 log 8x =. ln(x + ) + ln(x) = ln(x + ) 4. log x = ln x = ln (x + 4) + ln (x ) = ln 7 7. log (x 7) = - log x 8. log x + log (x + 15) = 9. 4 ln x = 11 10) Describe the transformation of f(x) = ln x from the function g (x) = ln x. 11) The population of deer in a forest preserve can be modeled by the equation P = ln (t + 1), where t is the time in years from the present. In how many years will the deer population reach 1000? 1) Simplify: c 5 c 10 c 4 1) Simplify: 16b 5 c d Condense the expression to a single logarithm: log 4 5 log 4 y + log 4 x ) 15. Give an exact solution for the following equation: x = x 16. Solve for x: log 9 (x + 7) + log 9 x = log 9 (x) + log 9 (x + 5) 17. Solve: log 4 (x + ) = log 4 (x ) 7.8 Practice Problems 1. Miguel and Dee determined slightly different equations to model the recommended chair seat height, in inches, for children x years old. (Miguel: y = 4.7 ln x ; Dee: y = 4.5 ln x ) For what age do the two models give the same chair seat height?. Latrell wants to double an investment of $,500 that earns interest at an annual rate of 6% compounded continuously. Which equation can he solve to find the doubling time t for this investment? A = 500 ln 0.06t B = 500e 0.06t C = 500(1.06) t D = 500 log 0.06 t

8 8. The half-life of cesium-17 is 0 years. Which equation gives the time t (in years) needed to reach a percent p of remaining radioactive substance? A. t = log1( p ) B. t = 10 log1 ( 10 ) p log1 ( p C. t = 100 ) 0 D. t = 0 log1( p ) The formula L = 10(logI + 1) gives the sound level L (in decibels) for a sound with intensity I (in watts/m ). What is the intensity of a sound whose sound level is 14 decibels? 5. Scientists have determined that the population for a particular species in a habitat can be modeled by the equation P = (500t 180) 6 7. How many years (t) will it take the species to grow to 79 members? 6. A person wants to invest $4800 in a savings account for 7 years. The bank offers the two options below. Explain which option the person should choose. Option A Interest Compounded Monthly A = P (1 + r nt n ) Rate:.7% Option B Interest Compounded Continuously A = Pe rt Rate:.65% 7. Which of the following functions would NOT produce the same graph as (x) = 56 x? A. h(x) = 8x B. h(x) = 4 4x C. h(x) = 8 x D. h(x) = 16 x 8. Write an exponential function in the form y = ab x to model the set of data given in the data below. x ln y Three people in the business club are competing to see who can double their investment in the shortest amount of time. Each person starts with an initial amount of $000, but they each choose different investment scenarios. Who will double their investment first based on the following information? Person A Person B Person C Interest compounded quarterly A = P (1 + r n ) nt Rate: 6.% Interest compounded daily A = P (1 + r n ) nt Rate: 5.9% Interest compounded continuously A = Pe rt Rate: 5.7%

9 9 Practice Test Part 1. How long would it take for $1000 deposited in an account paying 6% interest compounded continuously to earn $00 interest?. How much money must be deposited now in an account paying 8% annual interest, compounded quarterly, to have a balance of $1000 after 10 years?. How much money must be deposited now in an account paying 7% annual interest, compounded continually, to have a balance of $1000 after 6 years? 4. How long will it take $500 to double if it earns interest that is compounded monthly at a rate of.8%? For #5 15: Solve for x. Round to the nearest hundredth, if needed. 5. x log log 8 x 7. log log x = 9. ln (x ) = ln ( x ) log ( x 1) log 10 x log x x 1 1. e ln x = 7x x = 6 x 16. Condense the expression: log 7 x log 7 y + 4log 7 z 17. Expand the expression: ln 1x4 5y Condense the expression: ln ln x ln y + 1 ln z Use the change-of-base formula to evaluate the expression. Write the exact answer and an approximate solution to decimals. 19. log 9 0. log 5 1. Suppose that the magnitude of an earthquake measures 7.5 on the Richter scale. Find the amount of energy E released by this earthquake m = / log (E/ ) Solve each equation. If needed, round the nearest hundredth.. 5 x + 4 = e 0.07t = log x = ln x = 0 6. ln (x + ) + ln (x 5) = ln 7. 9e -4x = x+1 = 5 x+ 9. log x + log (x 6) = 4 0. log 4 (x ) = 1 log 4 x

10 10 Unit 7 Practice ANSWERS: 7.1 Answers 1... Up reflect over y =, up, left 1 right D: (, ); R: (, ) D: (, ); R: (-,) D: (, ); R: (0, ) as x, y as x, y as x, y 0 as x, y as x, y as x, y 4. $ $ $ $ a) 9,816,591 b)1.018, 1.8% c) 7,01,55 9. Decay 10. Decay 11. Growth 1. reflect over x-axis; vertical stretch by factor of ; left 1 1. Right, up 4 D: (, ); R: (-,0) D: (, ); R: (4, ) as x, y as x, y as x, y 0 as x, y D 15. B 16. B 17. 5x(x + )(x ) 18. (x + )(x )(x + )(x ) 19. (x + )(x 1) x 8. B 7. Answers 1. a) 4e 6 b) a) 1 7 b) e e. a) 8 b) e 5. 15e 8x+5

11 11 6. vertical compression by a factor of, right, down 7. Reflect over y =, vertical stretch by factor of, left 1, up D: (, ); R: (-, ) D: (, ); R: (-,) 8. $ $ a) $ b) $ z z x 15. (x 4)(9x + 1x + 16) 16. (x + i)(x i)(x + )(x ) 17. (x 11)(x + 5) 18. Not factorable (prime) 19. $ x = 7 1. x = 9. B 7. Answers 1. 4 = 81. log 4 1 = = log 1/55 = ½ / x 14. x 15. x 16. 4x $ x x 4. Charles should use Bank #1. After 4 years investing in Bank #1, Charles would have $17,181.9, but after investing in Bank # for the same amount of time, Charles would have only $17,048.0, which is a smaller amount of money. 5. II and IV 6. J 7.4 Answers 1. y = 7 x. y = e (x+) + 1. y = (x+)/ 4. y = e (x ) 5. left 1; D: (-1, ) ; R: (-, ) 6. Reflection in y = 0, vertical stretch by a factor of, left 1, D: (-1, ) ; R: (-, ) 7. right, up 1; D: (, ) ; R: (-, ) 8. Vertical compression by a factor of, down ; D: (0, ) ; R: (-, )

12 1 9. reflection over y = 0, right reflection over y =, vertical stretch by a factor of, D: [5, ) ; R: (-, 0] right 1, up D: [1, ); R: (-, ] 11. left, up 4 1. reflection over y = 5, vertical stretch by a factor of, up 5 D: (-, ) R: (-, ) D: (-, ) R: (-, ) 1) 5 = x; 5 14) x = 7; 15) e o = x; 1 16) log 416= x; 17) C 18) -7 19) A 0. left 1 1. Right, up D: (-1, ) ; R: (-, ) D: (, ) ; R: (-, ). Right. Reflection in y =, vertical stretch by factor of, right 1, up D: (, ) ; R: (-, ) D: (1, ) ; R: (-, ) 4. F

13 1 Test Part 1 Practice Answers 1. right, up 5; D: (-, ); R: (5, ). Reflect in y =, vertical stretch by a factor of, left 1, down ; D: (-, ); R: (-, -). right, up 4; D: (-, ); R: (4, ) 4. Left 1, down 6; D: (-1, ); R: (-, ) 5. Reflection in y = 0, vertical compression by a factor of ; D: (0, ) ; R: (-, ) 6. $ $ = log 100 = x y = e x 19. y = log 4x 0. y = 6 x Growth; decay 7.5 Answers log x 6. log log 5 x log 5 - log 5 y 7. ½ ln x + ½ ln y 8. ln ln x 9. log log 6 x log 6 y 10. log x1/ ln x 1. ln (x+1) y 1. log 4x y 14. log log 1 log 7 ; log 1.5 log 5 ; log log. log 4 ; ; $ $ $409.5 log , 19

14 Answers ; ln ; ln5 ln ;log (4) ; ln 5 ln years years 1. $ No 1. D 7.7 Answers 1. 1/ ± 4. 1,000, ) 5 9) ) Down units, reflection in the x-axis, vertical stretch by a factor of 11) years 1) c c 1) bd 4 b c 14. log 4 15x y 15. log 48 log No solution Answers 1. about.1 years old. B. D 4..5 watts/m years 6. Option A 7. C 8. y =.5() x 9. Person A Practice Test Part Answers years. $ $ years ± log 7 4x z 4 y 17. ln ln x ln 5 6 ln y 18. ln x4 z y 19. log 9 log log 5 ; ; x 10 log