Solutions Key Exponential and Logarithmic Functions

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1 CHAPTER 7 Solutions Key Exponential and Logarithmic Functions ARE YOU READY? PAGE D. C. E. A. x ( x )(x) = x (x) = x 6 6. y -1 ( x y ) = ( y -1 y ) x = (y) x = 1 x y 7. a 8 = a (8 - ) a = a 6 9. x y = x ( - 1) y ( - 6) x y 6 = x 1 y -1 = x y 11. (x) ( x ) = 9 x ( x ) = 6 x 1. a - b = a (- - ) b - (-1) a b -1 = a -6 b = b a 6 1. I = 000(%)() = 180 The simple interest is $ (r)() = r = 90 r = 0.01 or 1.% The interest rate is 1.%. 1. P + P(6%)() = P + P(0.18) = 1.18P = P = 00 The loan is $ x - y = x = y + x = y x = y - x = (y - ) x = 6y y 1 y (1 - ) = y = y. ( x ) - = ( ( x ) -1 ) = ( x ) = 7 x 17. y = -7x + y + 7x = 7x = -y + x = -y y = x - y = x - -x = -y - x = y , , EXPONENTIAL FUNCTIONS, GROWTH, AND DECAY, PAGES CHECK IT OUT! PAGES growth. P(t) = 0(1.1 ) t The population will reach 0,000 in about 0.9 yr.. v(t) = 00(0.8 ) t The value will fall below $0 in about 1. yr. Copyright by Holt, Rinehart and Winston. 7 Holt Algebra

2 THINK AND DISCUSS, PAGE ; ; Possible answer: although 1.01 and 0.99 are very close to 1, gets very large because it is a growth function, and gets close to 0 because it is a decay function.. Possible answer: f(x) = 1. 1 x shows growth, and f(x) = 0. 9 x shows decay. The graphs intersect at (0, 1).. Possible answer: exponential decay; exponential growth. c. The number of bacteria after 1 hours will be about 600,000. 6a. f(x) = (0. ) x b. c. A new softball will rebound less than 1 inch after bounces. PRACTICE AND PROBLEM SOLVING, PAGES decay 8. growth EXERCISES, PAGES 9-96 GUIDED PRACTICE, PAGE 9 1. exponential decay. decay. growth 9. growth. decay a. f(x) = 10 ( x ) b. a. f(t) = 80(1.0 ) t b. c. The number of ton-miles would have exceeded or would exceed 1 trillion in year, or a. f(x) = (0.9) x Copyright by Holt, Rinehart and Winston. 76 Holt Algebra

3 b. c. About 6 units will remain after minutes. d. It will take about 1.6 min for half of the dose to remain. 1. No; the variable does not contain an exponent. 1. No; 0 x is 0, a constant function. 1. Yes; the variable is in the exponent = 8 (1 +.% ) 8 = (1.0 ) 8 = 1,9,9.1 The balance in 008 will be about $1,000,000. t 16. N(t) =.( ) 17. Let x be the number of years needed. 76(1-0% ) x = 0 76 ( 0.7 x ) = x x.8 It will take about.8 years for the computer s value to be less than $ ; 0.1; 0.; 1.00;.0;.8;.6;.; ; 6.;.0; 1.00; 0.0; 0.16; 0.06; 0.0; a. b. You will owe $119.6 after one year. c. It will take about 18 months for the total amount to reach $0. 1a. 00(1-0% ) 6 = 00(0.8 ) 6 16 The rep sold about 16 animals in the 6th month after the peak. b. Let x be the number of month that the rep first sold less than 00 animals. 00(1-0% ) x = ( 0. 8 x ) = x 0.08 x 1 The rep first sold less than 00 animals in the 1th month. a. A = P ( 1 + r n) nt = 000 ( 1 + % ) = 000(1.01 ) 0 = 6.19 The investment will be worth $6.19 after years. b. Let x be the number of years needed. 000 ( 1.01 x ) = x = x 1 The investment will be worth more than $,000 after 1 years. c. A = P ( 1 + r n) nt = 000 ( 1 + % 1 ) 1. (0, 1) = 000( ) = 6.60 His investment will be worth $6.60 more after years.. (0, 8,0]. (.868, 0] 6., 768 (00-1) 7a. = 8 = 0.17 = 17% There is a 17% decrease in the amount each day. b. A(t) = 00(1-17% ) t = 00(0.8 ) t c. A(1) = 00(0.8 ) About 6.8 mg will remain after 1 days. 8. N(t) = 6.1(1 + 1.% ) t = 6.1(1.01 ) t t = = 0 N(0) = 6.1(1.01 ) The population will be about 8.1 billion in x ; when x =, they are equal, but x becomes greater quickly as x increases. 0. Possible answer: A company doubles in size each year from an initial size of 1 people; f(p) = 1( ) x ; f() = 1( ) means there are 96 people in yr. TEST PREP, PAGES B. H. a = 1, b =.. B CHALLENGE AND EXTEND, PAGE 96. The degree of a polynomial is the greatest exponent, but exponential functions have variable exponents that may be infinitely large. Copyright by Holt, Rinehart and Winston. 77 Holt Algebra

4 6. x < x <. 7. x > Let x be the cost of a new video game, and y be the cost of an old one. x + y = x + y = 19 Solve equation 1 and equation for x and y using elimination. (x + y) = () 6x + y = 70 -(x + y) = -19 -x - y = -19 x = 7 x = Substitute for x into equation to find y. x + y = 19 () + y = 19 y = 10 y = The cost of a new video game is $, and the cost of an old one is $. 7. odd; positive; 1 8. even; positive; 9. even; negative; 9. ; (, ), (-0.767, 0.88) 0. (0) + = = 0 d + = () d + = 0 d +.6 d 1. There are 0 mosquitoes per acre at the time of the frost; it takes about 1. days for the population to quadruple. 1. If b = 0, f(x) = 0; if b = 1, f(x) = 1. These are constant functions. If b < 0, noninteger exponents are not defined. SPIRAL REVIEW, PAGE 96. D: {x x }; use y; f(x) = x shifted right units. D: ; R: ; f(x) = x stretched vertically by a factor of. D: ; R: {y y 1}; f(x) = x reflected across x-axis and shifted 1 unit up. D: ; R: ; f(x) = x shifted units down TECHNOLOGY LAB: EXPLORE INVERSES OF FUNCTIONS, PAGE 97 TRY THIS, PAGE D: ; R: {x x 0};. D: ; R: for both; inverse. yes; for any input, there D: {x x 0}; R: ; is at most 1 output. no; it fails the vertical line test for {x x > 0}.. The domain of a function is the range of the inverse, and the range of a function is the domain of the inverse.. They are reflections across f(x) = x. 7- INVERSES OF RELATIONS AND FUNCTIONS, PAGES 98-0 CHECK IT OUT! PAGES relation: D: {1 x 6}; R: {0 y } inverse: D: {0 y }; R: {1 x 6} a. f -1 (x) = x b. f -1 (x) = x -. f -1 (x) = x + 7 Copyright by Holt, Rinehart and Winston. 78 Holt Algebra

5 . f -1 (x) = x - 8. f -1 (x) = (x + 1) 9. f -1 (x) = (x - ). f -1 (x) = -x f -1 (x) = - x f -1 (x) = x f -1 (x) = x + 1. f -1 (x) = - x + 1. f -1 (x) = (x - ). inverse: z = 6t - 6 = 6(7) - 6 = 6 6 oz of water are needed if 7 teaspoons of tea are used. THINK AND DISCUSS, PAGE Possible answer: When x and y are interchanged, the inverse function is the same as the original function. The graph of an inverse is the reflection across y = x, but the original function is y = x.. Possible answer: y = x; y = x. Possible answer: You get the original function; yes, the original is a function f -1 (x) = x F = 9 C + = 9 (16) + = C is about 61 F. EXERCISES, PAGES 01-0 GUIDED PRACTICE, PAGE relation. relation: D: {1 x }; R: {1 y 8}; inverse: D: {1 x 8}; R: {1 y };. relation: D: {-1 x }; R: {- y -1}; inverse: D: {- y -1}; R: {-1 x }; PRACTICE AND PROBLEM SOLVING, PAGES relation: D: {-1 x }; R: {1 y } 19. relation: D: {- x }; R: {- y } inverse: D: {1 y }; R: {-1 x } inverse: D: {- y }; R: {- x } 0. f -1 (x) = 1. 1 x 1. f -1 (x) = x + 1. f -1 (x) = 0.x. f -1 (x) = - x + 1. f -1 (x) = 0.08x f -1 (x) = x f -1 (x) = x -. f -1 (x) = x 6. f -1 (x) = x 7. f -1 (x) = x + Copyright by Holt, Rinehart and Winston. 79 Holt Algebra

6 6. f -1 (x) = x f -1 (x) = 1.1 x 7. f -1 (x) = -x f(x) = 1900x , where x is the number of years after 001; f -1 (x) = x , where x is the number of 1900 bachelor s degrees awarded; f -1 ( 1.7 6) = About years after 001, 1.7 million bachelor s degrees will be awarded. 0a. m = y - y 1 x - x = = - b. The slope of the inverse line is the negative recipical of the slope of the original line, which in this case is - ( -) = -. 1a. f -1 (x) = 1 - x 1.8 b. f -1 (00) = = 600 The boiling point of water will fall below 00 F above about 600 ft. c. f -1 (160.) = = = 796 The mountain peak s altitude is 7,96 ft.. (, -), (1, ), and (-, -). (, ), (, ), (-, -1), and (-1, -). The inverse is x =, which is a line parallel to the y-axis. So it is not a function.. f(x) = 1.9 x; f -1 (x) = 1.9 x = 1.9x f -1 () = 1.9() = 1.8 It will take Warhol 1.8 s to complete a m race. 6a. C = n +. b. n = C -. = = 7 Seven tickets are purchased when the credit card bill is $17.0. c. n =.0 -. = 1.9 No; when C = $.0, n is not an integer. 7. B; the student may have found the inverse of each term. 8. The function is inversed, and the graph is reflected over the line y = x. The result may or may not be a function. 9. Yes; possible answer: for the ordered pairs (, 1) and (, ), the inverse relation is (1, ) and (, ), which is a function. 0a. S(c) = c - 7 s + 7 b. C(s) = = ( s + ) 7 = s ; yes; head circumference as a function of hat size. c. C ( 7 8) = ( 7 8) = The head circumference of the owner is in. 1. always. sometimes. never. always. always 6. always 7a. P = 17 0 d b. D: {d d 0}; R: {P P 1.7} c. d = P = 0 (P - 1.7) = P - ; 0 depth as a function of pressure d. At.9 ft, the pressure is.9 psi. TEST PREP, PAGES A; x + f -1 (x) = = ( x + ) = x F 0. C 1. x 1 y 0 1 CHALLENGE AND EXTEND, PAGE 0 (x - b). y = m = x m - b m. ay + bx = c ay = c - bx y = c - bx a y = - b a x + c a. x - y 1 = m(y - x 1 ) x - y 1 = my - m x 1 x - y 1 + m x 1 = my x - y 1 + m x 1 = y m x - y 1 m + x 1 = y Copyright by Holt, Rinehart and Winston. 80 Holt Algebra

7 . a. log = - b. lo g 0.0 = -1. D: {x x > 0}; R: y = x ; switch x and y: x = y 6. Either the function and its inverse are both f(x) = f -1 (x) = x, or the function and its inverse are both f(x) = f -1 (x) = -x + k, where k is any real number constant SPIRAL REVIEW, PAGE a..9,.18, 6.0, 6.89, 7., 8.16 b. {v.9 v 8.16} 61. (x + )(x - )(x - 1) = 0 ( x + x - 6) (x - 1) = 0 ( x - 7x + 6) = 0 x - 1x + 1 = 0 6. [x - (1 - i)][x - (1 + i)](x - ) = 0 [(x - 1) + i][(x - 1) - i](x - ) = 0 ((x - 1 ) - i )(x - ) = 0 ( x - x (-1)) (x - ) = 0 ( x - x + ) (x - ) = 0 ( x - x + 6x - ) = 0 x - 8 x + 1x - 8 = 0 6. (x + )(x - 8)(x - 9) = 0 ( x - x - ) (x - 9) = 0 ( x - 1 x + 1x + 16) = 0 x - 8 x + x + = 0 6. decay 66. decay 67. growth 68. growth 7- LOGARITHMIC FUNCTIONS, PAGES 0-11 CHECK IT OUT! PAGES a. lo g 9 81 = b. lo g 7 = c. lo g x 1 = 0 a. 1 = b. 1 = 1 c. ( ) - = 8 6. (x - ) (x + ) = 0 ( x - ( ) ) = 0 x - = 0. ph = -log ( ) =.8 The ph of the iced tea is.8. THINK AND DISCUSS, PAGE Possible answer: If x = log b b, then b x = b, or b x = b 1, so x = 1.. Possible answer: no; lo g 16 =, but lo g 16 = 0... EXERCISES, PAGES GUIDED PRACTICE, PAGE x. lo g. 1 = 0. lo g 8 = 1.. log 0.01 = -. lo g = x 6. - = x = = = x. lo g 7 = 11. lo g ( 9) = - 1. lo g = 1. lo g = 1. D: {x x > 0}; R: 1. D: {x x >0}; R: 16. poh = -log ( ) = 8. The poh of the water is 8.. Copyright by Holt, Rinehart and Winston. 81 Holt Algebra

8 PRACTICE AND PROBLEM SOLVING, PAGES lo g x =. 18. lo g 6 (16) = x 19. lo g 1. 1 = 0 0. lo g 0. = = 6. 6 = x.. 0 = 1. π 1 = π. lo g 1 = 0 6. log = - 7. lo g 6 = 8. lo g = - 9. D: {x x > 0}; R: 0. D: {x x > 0}; R: 1. ph = -log ( ) = 6. No; the ph is 6., so the flowers will be pink. log A - log P a. n = = log log 00 = 11 log (1.017) log (1.017) The debt has been building for 11 months. log A - log P b. n = = log 10 - log 1.6 = 9 log (1.017) log (1.017) It will take 9 additional months until the debt exceeds $10. c. As the debt builds, it takes a shorter time for the debt to increase by a relatively constant amount.. 1; lo g a b = 0 means a 0 = b and a 0 is 1 for any a 0. = log ( 1 ) = (1) a. Jet: L = log ( 1 ) = 10 db; Jackhammer: L = log ( 1 ) = log ( 1) = (1) = db; Hair dryer: L = log ( 7 ) = (7) = 70 db; Whisper: L = log ( ) = 0 db; = log ( 7) = log ( ) = () = log ( ) Leaves rustling: L = log ( ) = () = 0 db; Softest audible sound: L = log ( ) = log (1) = (0) = 0 db b. Let the intensity of the rock concert be x. log ( x ) = 1 log (x) = 1 log (x) = 11 x = 11 The intensity of the rock concert is 11, and it should be placed between jackhammer and hair dryer. c. No; 0 bels is 00 db.. Yes; possible answer: log 1 0 = log 00 =, and there are zeros after the 1. log = log 0.01 = -, and there are zeros log 00.; log a. orange b. lemon c. grapefruit 8. lo g 0 means 0 x =, and lo g 1 means 1 x =. These have no solution. TEST PREP, PAGES C 0. G 1. A. F. 6 = 6 lo g 6 = 6 CHALLENGE AND EXTEND, PAGE 11. The range of lo g 7 x is negative for 0 < x < 1 and positive for x > 1. The range of lo g 0.7 x is positive for 0 < x < 1 and negative for x > 1.. lo g 9 = ; lo g 7 = ; lo g = lo g 9 + lo g 7 = lo g lo g b ( b x ) + lo g b ( b y ) = lo g b ( b x + y ) 6. Let lo g 7 7 x + 1 = a. Write the above exponental equation as a logarithmic equation, we obtain 7 a = 7 x + 1 a = x + 1 Hence we have proved that lo g 7 7 x + 1 = x a. 11 = 08 Hz, lo g 08 = 11 b. octaves lower; lo g 6 = 8, lo g =, 8 - = Copyright by Holt, Rinehart and Winston. 8 Holt Algebra

9 SPIRAL REVIEW, PAGE ( ( a ) ( b ) ) ( a b ) 1 0 ( a ) ( b ) 0 a 8 b 0. - t ( s t -1) (- )s ( t t -1 ) -st 1. 7 a - b (ab + a -1 b ) 9. 8 s t 6 s t 8 8 s s t 6 t 8 s s 7 a - b ab + 7 a - b a -1 b 1 a -1 b + 8 a - b. 0 = - 16 t 16 t = t = 16 t = 16 t = It took s or 1. s for the brick to hit the ground.. 0., 0.9, 1, 1.7, , 1.67, 1, 0.6, ,., 1, 0., PROPERTIES OF LOGARITHMS, PAGES 1-19 CHECK IT OUT! PAGES 1-1 1a. lo g 6 + lo g lo g (6 ) lo g lo g lo g 1 7 lo g 7 ( 9 7 ) lo g b. lo g lo g () t t b. lo g 7 + lo g 9 lo g 7 ( lo g -1 a. log 1 0 log (1) a. 0.9 b. 8x a. lo g 9 7 lo g 7 lo g 9 1. c. lo g ( ) ) lo g ( (-1) - b. lo g 8 16 lo g 16 lo g ) 6. 8 = log E ( ( ) 11.8 ) 8 = log E ( ) = log ( E ) = log E - log = log E = log E.8 = E.6.8 = About 6 times as much energy is released by an earthquake with a magnitude of 9. than by one with a magnitude of 8. THINK AND DISCUSS, PAGE 1 1. Change the base and enter Y=log(X)/log() is , and is about.98, so is about You get lo g b a =. lo g a b. EXERCISES, PAGES GUIDED PRACTICE, PAGE lo g 0 + lo g 6. lo g (0 6.) lo g 1. lo g + lo g 7 lo g ( 7) lo g 81. log 0 + log 00 log (0 00) log lo g 0 - lo g lo g ( 0 ) lo g 6. log. - log 0.0 log ( 6. lo g lo g ). lo g 6 ( ) log 0 lo g Copyright by Holt, Rinehart and Winston. 8 Holt Algebra

10 9. lo g 7 9 lo g 7 ( 7 ) lo g x lo g lo g 1. lo g 9 7 lo g 7 lo g lo g log log = log E ( ( ) 11.8 ) 8.1 = log E ( ) = log ( E ) = log E - log = log E = log E.9 = E. lo g (0.) lo g ( 0. ) lo g lo g (0. ) lo g ( -1 ) lo g lo g 8 lo g lo g lo g 7 log 7 log = log E ( ) 11.8 ( ) 7.9 = log E ( ) = log ( E ) = log E - log = log E = log E.6 = E = About times as much energy was released by the 1811 earthquake than by the 197 one. PRACTICE AND PROBLEM SOLVING, PAGES lo g 8 + lo g 8 16 lo g 8 ( 16) lo g 8 6. lo g..1 + lo g. lo g. (.1 ) lo g lo g 16 - lo g lo g ( 16 ) lo g 8 6. lo g 1 6 lo g ( ) lo g log + log log ( ) log 1. log 00 - log 0 log ( 00 0 ) log 1. lo g lo g 1. lo g 1. ( 6.7 ) lo g log (0 ) 0.1 log ( ) 0.1 log lo g 1 lo g ( ) lo g x lo g lo g 9 9. lo g 16 lo g 16 lo g lo g 9 log 9 log = log ( I ) = log ( I ) = I = I. lo g 1 lo g 1 lo g = log ( I. = log ( I 1 0. = I 1 0. = I ) ) = The concert sound is bout.16 times more intense than the allowable level. 6a. 1 - (-.) = log d 6. = log d 1.6 = log d = d 18. d 18 = d The distance of Antares from Earth is about 18 parsecs. b..9 - M = log.9 - M = log..9 - M (1.).9 - M = 6.7 M = -.8 The absolute magnitude is about -.9. Copyright by Holt, Rinehart and Winston. 8 Holt Algebra

11 c. - (-0.) = log d. = log d 1.08 = log d = d 1.0 d = d The distance to Antares is about 1. times as great as the distance to Rho Oph. 7. lo g b m + lo g b n = lo g b mn 8. lo g b m - lo g b n = lo g b m n 9. lo g b ( b m ) n = lo g b mn n lo g b b m = mn 0. lo g - lo g 18 lo g ( 18) lo g -. - lo g lo g lo g lo g (1) log log 1 + log log (0.1 1 ) log 1 0. lo g + lo g lo g ( ) lo g - log. log (1) 1 6a. log 0 = log ( ) = log + log b. log 00 = log ( ) = log + log c. log 000 = log ( ) = log + log a. P = 1(1 - % ) t = 1(0.96 ) t 1) b. t = lo g 0.96 ( P c. log ( 1) x log 0.96 log ( 1) 1) 0 = log The population will drop below 0 after about d. t = lo g 0.96 ( 0 9 years. 9. P = 0(1 + 8% ) t = 0(1.08 ) t t = lo g 1.08 ( 0) P = lo g 1.08 ( 0 log ( 0) 0 0) = log It will take about.9 years for the value of the stock to reach $0. 0a. (00) = 00(1.016 ) n = (1.016) n log = log (1.016) n log = n log (1.016) log log (1.016) = n.7 n It will take about.7 months for the debt to double. b. It will take another.7 months for the debt to double again. c. no Possible answer: Change the base from b to by writing log b x as log x. Enter log(x)/log(b), using log b the calculator s LOG key. a. lo g = lo g 1 ( 0) = lo g 1 - lo g b. lo g = lo g 1 (0 ) = lo g lo g c. lo g 1 00 = lo g 1 0 = lo g 1 0 (1.). 6a. log b. log ( ) = log. + log c. log (. 1 0 ) = log. + log log ( a 1 0 x ) = x + log a d. log ( ) = log. + log Yes, the conjecture holds for scientific notation with negative exponents. 7. sometimes 8. always 9. always 60. never Copyright by Holt, Rinehart and Winston. 8 Holt Algebra

12 61. always 6. never 6. sometimes 6. never 6. B; log 80 + log 0 log (80 + 0) TEST PREP, PAGE B 67. H; lo g 9 x + lo g 9 x lo g 9 x + lo g 9 x lo g 9 x CHALLENGE AND EXTEND, PAGE A; log 6 log ( ) log + log 69a. Possible answer: the on the top scale is lined up with the 1 on the lower scale. At on the lower scale, the product, 6, is read on the top scale. So the sum of log units and log units is log 6 units. b. The lengths show log + log = log( ) = log {x x < - x > } 71. {x x > 1} 7. ( x > 1 and x > 0) or 7. {x x > 0} ( x < 1 and x < 0) {x -1 < x < 0 x > 1} 7. {x x > -1} 7. {x -1 x < 0} 76. Let lo g b a p = m b m = a p Let lo g b a = n b n = a Substitute b n for a into equation 1. b m = ( b n ) p b m = b np m = np Substitute lo g b a p for m and lo g b a for n into equation. lo g b a p = p lo g b a 77. lo g 9 x lo g 9 ( ) x x lo g 9 9 x(1) x 78. x = x = x = 79. x = x 0 = 1 x = -8 The solution for the x = - equation is Since x > 0, there is no {x x > 0 and x 1}. solution for the equation. SPIRAL REVIEW, PAGE = (x - 1) 9 = x - -x = -1 x = n = n + 9 7n = 9 n = ( ()(i) 1i 16 ) ( -1 ) 8. (x + 1) = (x - 6) x + = 6x x = - x = ( n + ) = n - 8n + 6 = n - -n = - n = ( 0 )( -1) - ( )(i) -i ( 8 ) ( ( )(i) 8i -1 ) ( 1 )( -1) ( )(i) i 89. lo g 1 = 90. log 0.1 = lo g 6 6 = lo g 6 = x = 1 lo g 1 1 = = lo g 16 = = lo g = = 0.0 lo g = -0. MULTI-STEP TEST PREP, PAGE M = ( 18.% 1 ) 1 - ( % ) -1 The monthly payment will be about $ = You will end up paying $1,187.0 in five years. log ( (18.%) 160(1) ). t = -1log ( % 9. 1 ) It will take about 9. years to pay off the debt % = No; the monthly interest alone amounts to $11.1, so the payments do not even keep pace with the interest.. $11.16 READY TO GO ON? PAGE 1 1. decay. growth a. p = 00(1. ) d. decay. growth Copyright by Holt, Rinehart and Winston. 86 Holt Algebra

13 b. There will be about 17,086 bacteria in the culture the following Monday f -1 (x) = x -.1. f -1 (x) = x f -1 (x - ) (x) = f -1 (0.0 - ) (0.0) = 9. f -1 (x) = ( ) - x 11. f -1 (x) = x - = 19.0 The number of hours of labor is. h. 1. lo g 9 = 1. lo g = 0 =. 1. lo g 0. = lo g = x 17. = ( ) - = = 1 0. e = x 1.. lo g 81 + lo g 9 lo g (81 9) lo g lo g lo g lo g - + lo g ( ) 1. lo g lo g lo g 1. ( ) lo g lo g 7 lo g lo g 7 lo g lo g 7 0. lo g 6 lo g. lo g 6 lo g 6 () lo g 0.01 lo g EXPONENTIAL AND LOGARITHMIC EQUATIONS AND INEQUALITIES, PAGES -8 CHECK IT OUT! PAGES - 1a. x = 7 x = x = x = 1. c. x = 1 log x = log 1 x log = log 1 log 1 x = log 1.0. n log n - 1 log 8 (n - 1)log 8 n log n b. 7 -x = 1 log 7 -x = log 1 -x log 7 = log 1 -log 1 x = log log + 1 n 18 You would receive at least a million dollars on day 18. a. = log 8 + log x = log 8 + log x = log 8 x = log 8 x = 8 x 1 = x = x b. log x - log = 0 log x = log log x = log x = x = Copyright by Holt, Rinehart and Winston. 87 Holt Algebra

14 a. x = b. x < c. x = 00 THINK AND DISCUSS, PAGE 1. If log a = log b, then 1 0 a = 1 0 b. The exponents must be equal for the exponential expressions to be equal. a. Write log x = as log x. b. Write log x + log as log x c. Take the log of both sides. d. Use as a base for each side. e. Rewrite log (x + ) + log (x) as log ( x + x). f. Use 6 as a base for each side.. Yes; possible answer: log (-x) = 1 has - as a solution.. EXERCISES, PAGES 6-8 GUIDED PRACTICE, PAGE 6 1. exponential equation. x = ( ) x = ( ) x = x = x = 8. x = x + 1 x = ( ) x + 1 x = x + x = x + x = - 6. ( ) x = ( ) x (( ) ) x = ( ) x ( ) x = ( ) x x = x x = 0. 9 x = x - ( ) x = x - x = x - x = x - x = -. x = log x = log x log = 1 x = log x + 1 = 9 log. x + 1 = log 9 (x + 1)log. = log 9 x + 1 = log 9 log. x = ( log 9 log. ) - 1 x = 00( ) t t log.9 log 1.0 t log.9 t log 1.0 log.9 log 1.0 t 6.8 t It will take 7 years for the population to exceed,000 people. 9. lo g (7x + 1) = lo g ( - x) 7x + 1 = - x 8x = 1 x = 8. lo g 6 (x + ) = 6 lo g 6 (x + ) = 6 x + = 16 x = 1 x = log 7 - log ( x ) = 0 log 7 = log ( x ) 7 = x 16 = x 8 = x Copyright by Holt, Rinehart and Winston. 88 Holt Algebra

15 1. lo g x 9 = 1 9 lo g x = 1 lo g x = x = x. 1. lo g 7 ( - x) = lo g 7 ( x ) - x = x = 1x x = log x + log (x + 8) = 0 log x(x + 8) = 0 x(x + 8) = x + 8x - 0 = 0 (x - )(x + 0) = 0 x = 16. log ( x + ) + log x + 1 = 0 log x ( x + ) = -1 x ( x + ) = x =. 19. x = 0 x + 1. log 0 + log ( x ) = log x = x = 1 0 x = x + x - 1 = 0 (x - 1)(x +1) = 0 x = 18. x 0. x > x = PRACTICE AND PROBLEM SOLVING, PAGES x - 1 =. 6 ( ) x = 8 x - 1 x - 1 = -6 ( - ) x = ( ) x - 1 x - 1 = -6 -x = x - x = - -x = x - -x = - x = 0.6. ( ) x - x = 1 x ( -1 ) x - = ( ) - x = x - x = x = x x = 0.8 ) -x = 1.6 log ( ) -x = log 1.6. (. (1.) x - 1 = 1. log(1.) x - 1 = log 1. (x - 1)log 1. = log 1. log 1. x - 1 = log 1. log 1. x = log x 7.9 x = 1. x log + 1 = log 1. ( x + 1 ) log = log 1. x log = log x = log 1. ( log ) - 1 x > ( t 1 ) -x log = log 1.6 log 1.6 x = - log x > t 1 log 0.1 > log ( t 1 ) log 0.1 > t log 1 log 0.1 log > t 1 1log 0.1 log > t 0.8 > t It takes 1 min for the amount to drop below 0 mg. 8. lo g (7x) = lo g (x + 0.) 7x = x + 0. x = 0. x = lo g ( 1 + x ) = 1 + x = x = 1 x = 0 0. log x - log (1.) = log x 1. x =.1 = x = Copyright by Holt, Rinehart and Winston. 89 Holt Algebra

16 1. lo g x =. lo g x =. lo g x = 0.6 x = log x - log ( x 0) = x log ( x x 0 ) = x log 0 = x = x. - log x = log ( x 1) = log x + log ( x = log ( x = x 00 = x 0 = x. x = 6. x 1. ) 1). x < 1 7. log x = log ( x - 1) x = x = x - x = (x - )(x + ) x = or - log(-) is undefined, so the only solution is x =. 8. x = 0 log x = log 0 x log = x log = 1 x = log 1. 0a. = lo g l = l 8 = l 9. x + = 6 x + = 6 x + = 6 x = b. 0 = lo g l 0 = l 1 = l n = lo g (1) = lo g = -1 The f-stop setting is f/. n 1. 1 = 0 1 n = 1 n - = 1 - = n 1 - = n The position is keys below concert A.. 00 = 0(1 +.% ) n = (1.0) n log = log (1.0) n log = n log 1.0 log log 1.0 = n 1.7 n It will take at least 16 quarters or yr.. 0; log x = log x, no value of x satisfies the inequality; the graphs coincide, so there is no region where log x < log x.. The student solved log (x + ) = 8.. Method 1: Try to write them so that the bases are all the same. x = 8 x = ( ) x = 9 x = 9 Method : Take the logarithm of both sides. x = log x = log x log = 1 x = log 6a. Decreasing; is less than 1. b. t = = 0 N(0) = 119(0.987 ) 0 = 119(1) = 119 t = = 0 N(0) = 119(0.987 ) 0 9 There are 119,000 farms in 1980 and 9,000 in 000. c = 119(0.987 ) t 67.7 (0.987) t 0 t = 0 The number of farms will be about 80,000 in 0. 7a.. = 18( ) h h log h h.9 = 18( ) h h log h 11 h The lowest altitude is 11 km, and the highest is km; the model is useful in lower stratosphere and upper troposphere. b. P(0) = 18( ) (0) = 18( ) 0 = 18(1) = kpa = psi = 18.6 psi The model would predict a sea-level pressure greater than the actual one. Copyright by Holt, Rinehart and Winston. 90 Holt Algebra

17 TEST PREP, PAGE 8 8. B; b x = c log b x = log c x log b = log c x = log c log b 0. B CHALLENGE AND EXTEND, PAGE 8 9. J; log (x - 1) = - log x log (x - 1) + log x = log x(x - 1) = x(x - 1) = 1 0 x - 1x - 0 = 0 (x - )(x + ) = 0 x = 1. Possible answer: no; x = x x, and x 1 = x x, so x = 1, but lo g 1 is not defined.. x = 0.1 lo g x = ( - ) lo g x = -lo g x = lo g - x = - x = or lo g 6 - lo g x > 1 lo g ( 6 x ) > 1 6 x > 1 x < 1 {x x < 1} SPIRAL REVIEW, PAGE 8 a. 0.7x + 0.y. b. He can buy six -by-6-inch photographs.. deta = (7) - (1) = 6 6. deta = -1() - 9(-) = 7. deta = (6) - 0(-1) = 8. deta = (9) - (6) = 9. f -1 (x) = x - = x f -1 (x) = 6 x f -1 (x) = (x - 9) = x f -1 (x) = x = 7 x + 7 CONNECTING ALGEBRA TO PROBABILITY: EXPONENTS IN PROBABILITY, PAGE 9 TRY THIS, PAGE 9 1. P = ( ) 6 = = 1.6% 6. P = ( 6) 0.16 = 16.%. Let n be the number of free throws that the player makes. (70%) n < % log 0. 7 n < log 0.1 n log 0.7 < -1 n > -1 log 0.7 n > free throws will make the probability of making them all drop below %.. Let n be the number of questions guessed. (0.) n < 0.01% log (0.) n < log n log 0. < - n > - log 0. n > 6.6 The probability of guessing all of them correctly will drop below 0.01% for 7 questions. TECHNOLOGY LAB: EXPLORE THE RULE OF 7, PAGE 0 TRY THIS, PAGE 0 1. Rate A 0.0 A 0.0 A 0.0 A A Doubling Time B B B 1.77 B B Product C C C C C The lowest interest rate is 8.%.. The product of the interest rate written as a percent and the doubling time is about d = 0.7 r Copyright by Holt, Rinehart and Winston. 91 Holt Algebra

18 7-6 THE NATURAL BASE, e, PAGES 1-6 CHECK IT OUT! PAGES a. ln e. =. b. e ln x e ln x c. ln e x + y = x + y x. A = P e rt = 0 e 0.0(8) 1.1 The total amount is $1.1.. = 1 e -k(8) ln -8k = ln e ln -1 = -8k -ln = -8k k = ln = 60 e -0.08t = e -0.08t ln t = ln e 60 ln = -0.08t ln 00 t = I will take about 7.6 days to decay. THINK AND DISCUSS, PAGE 1. Possible answer: e and π are irrational constants. π is a ratio of parts of a circle and is greater than e.. e x and ln x are inverse functions. ln represents the logarithm, or exponent, when e is used as a base.. EXERCISES, PAGES -6 GUIDED PRACTICE, PAGE 1. f(x) = ln x; natural logarithm. 6. ln e 1 = 1 7. ln e x - y = x - y ( 8. ln e ) = - x 9. e ln x = x. e ln x = e ln x = x. 11. A = P e rt = 770 e 0.0() The total amount is $ = 1 e -k(6) ln -6k = ln e ln -1 = -6k -ln = -6k k = ln N() = 0 e -0.11() 16 The amount remaining after hours is about 16 mg. PRACTICE AND PROBLEM SOLVING, PAGES ln e 0 = ln e a = a 19. e ln (c + ) = c + 0. e ln x = e ln x = x 1. A = 000 e 0.0(). The total amount of his investment after years is about $.. Copyright by Holt, Rinehart and Winston. 9 Holt Algebra

19 . = 1 e -k(1) ln -1k = ln e ln -1 = -1k -ln = -1k k = ln N(000) = 0 e (000) 17 1 = 0 e t 0 = e t ln t = ln e 0 ln 0 = t ln t = The decay constant is ; the amount remaining is 17 g after 000 years; it takes 0,000 years for the 0 grams to decay to 1 gram. a. ln.0; log e 0.; they are reciprocals. b. ln = log loge = log e. log x = ln x by change of base, so ln ln (log x) = ln ( ln x ln ) = ln x a. 10 = (06-70) e -0.8t 10 = 16 e -0.8t = e -0.8t ln = -0.8t ln 10 t = It takes about. min for the coffee to reach its best temperature. b. 10 = (06-86) e -0.8t 10 = e -0.8t 10 = e -0.8t ln 10 = -0.8t ln 10 t = It takes about.8 min for the coffee to reach its best temperature. 6. c. room: 17. min; patio: 16.9 min Possible answer: They are the same. y = lo g 6 x is changed to base e by y = ln x and to base by ln 6 y = log x log B 8. A 9. C 0a. t = = = 700 e 0k = e 0k ln 7 7 = 0k ln 7 k = The growth factor k is about 0.1. b. t = 0-19 = 6 P(6) = 700 e 0.1(6) The population would have been about.6 million in ln + ln x = 1 ln x = 1 x = e 1 x = e 0.. ln + ln x = ln x = x = e x = e x = ± e x = ± e ±7. ln - ln x = ln x = x = e x = e 0.. ln x - = 0 ln x = ln x = 1 x = e Copyright by Holt, Rinehart and Winston. 9 Holt Algebra

20 . ln x - ln x = 0 ln x - ln x = 0 0 = 0 {x x > 0} 7a. 6. e ln x = 8 ln e ln x = ln 8 ln x = ln 8 x = 8 x = Solve the equation by trial and error to get n =. Possible answer: ; yes, at 18% interest it takes 17 periods. 6. D: ; R: 0 < y π b. : y = 0 and y = 1 c. Possible answer: The epidemic spreads slowly at first, then steadily, and then it tapers off slowly at the end. 8a. 1: f(x) = x ; : f(x) = e x ; : f(x) = x ; b. (0, 1) c. 0 = 0 = e 0 = 1 9. Possible answer: a little more; $00 at 8% interest compounded daily for 1 year: A = 00 ( ) 6 $8.8; compounded continuously: A = 00 e 0.08 $8.9 0a. t = = 0800 = 00 e k = e k ln 08 = k ln 08 k = b. t = = 0 N(0) = 00 e (0) 8000 There will be about 8,000 farms in 0. c. 179 = 9 e k = e k ln = k ln 179 k = A(t) = A 0 e kt = 9 e 0.006(0) 10 The average size will be about 10 acres in 0. TEST PREP, PAGE 6 1. C. J. A. Possible answer: ln ( x ) CHALLENGE AND EXTEND, PAGE 6. Let n be the number of periods needed in one year. ( n ) n e a. f(x) = ln (-x) b. f(x) = -ln x c. f(x) = -ln (-x) d. one asymptote: x = 0 SPIRAL REVIEW, PAGE 6 8a. g(x) = f(x) c. g(x) = f(x) b. g(x) = f(x + ) 9. g(x) = f(x) + = - x + x g(x) = f(x + ) = - x - x g(x) = -f(x) = x - x +. g(x) = f ( ) x = - x +. lo g 8 + lo g lo g 8 lo g x -. lo g 6 - lo g 1 lo g 6 1 lo g 6 Copyright by Holt, Rinehart and Winston. 9 Holt Algebra

21 . lo g - lo g 187 lo g 187 lo g ( 9) - 7. lo g lo g 8 8 lo g lo g lo g + lo g 1 lo g 1 lo g 8. log x - log x log x x log x 7-7 TRANSFORMING EXPONENTIAL AND LOGARITHMIC FUNCTIONS, PAGES 7- CHECK IT OUT! PAGES a.. x j(x) y = 0; j(x) = x ; translation units right ; y = 0; f(x) = x ; vertical compression by a factor of. b. ; y = 0; j(x) = x ; reflection across y-axis and vertical stretch by a factor of -; x = -1; f(x) = ln x translated 1 unit left, reflected across the x-axis, and translated units down; D: {x x > -1}. g(x) = log (x + ). 0 > -1 log (t + 1) > -1 log (t + 1) 17 < log (t + 1) 17 < t + 1 t > The average score will drop to 0 after 8,680 years, and it is not a reasonable answer. THINK AND DISCUSS, PAGE 1 1. Possible answer: {x x < 0}. Possible answer: The transformations of x and f(x) are the same for the functions f(x) + k, f(x - h), af(x), f ( b x ), -f(x), and f(-x).. Possible answer: f(x) = a x : vertical translations and reflections across the x-axis change the range; f(x) = lo g b x: horizontal translations and reflections across the y-axis change the domain; no.. EXERCISES, PAGES 1- GUIDED PRACTICE, PAGE 1 1. x g(x) y = ; translation units down; R: {y y > }. x h(x) y = 0; translation units down; R: {y y > -} Copyright by Holt, Rinehart and Winston. 9 Holt Algebra

22 . x j(x) y = 0; translation 1 unit left ; y = 0; vertical stretch by a factor of - ; y = 0; vertical compression by a factor of and reflection across the x-axis; R:{y y < 0} -1; y = 0; reflection across both axes; R: {y y < 0] ; y = 0; vertical compression by a factor of 7. -; y = 0; vertical stretch by a factor of and reflection across the x-axis; R: {y y < 0} ; y = 0; horizontal compression by a factor of 1. x = 0; vertical stretch by a factor of. 1. g(x) = ( x + ) 1. g(x) = log (x - 1) < 6 + ln (t + 1) 11 < ln (t + 1) 1 < ln (t + 1) 1 e < t > 8.1 x = 0; translation units left and vertical stretch by a factor of.; D: {x x > -} x = 0; reflection across the y-axis, vertical compression by a factor of, and translation 1. units up t > e The model is translated 1 unit left, stretched by a factor of, and translated 6 units up; the height will exceed 17 feet after about 9 years. PRACTICE AND PROBLEM SOLVING, PAGES x g(x) y = -1; translation 1 unit down; R: {y y > -1} x h(x) y = 0; translation units left x j(x) y = -1; translation 1 unit right and 1 unit down; R: {y y > -1} Copyright by Holt, Rinehart and Winston. 96 Holt Algebra

23 ; y = 0; vertical stretch by a factor of -0.; y = 0; vertical compression by a factor of 0. and reflection across the x-axis; R: {y y < 0} ; y = 0; vertical stretch by a factor of and reflection across the y-axis x = ; translation units right; D: {x x > } ; y = 0; vertical compression by a factor of 0. -1; y = 0; horizontal stretch by a factor of and reflection across the x-axis; R: {y y < 0} -; y = 0; vertical compression by a factor of and reflection across both axes; R: {y y < 0} x = -; translation units left; vertical compression by a factor of, and translation units down; D: {x x > -} 7. x = 0; vertical stretch by a factor of and reflection in the x-axis. ) x - 9. f(x) = ln (x + ) f(x) = -1. ( (1 - x 0. f(x) = e ) x = 0; vertical stretch by a factor of and reflection across the y-axis t e - 17 t e - 17 ln 60 t t -17ln The model is horizontally stretched by a factor of 17, reflected across the y-axis, and vertically stretched by a factor of 870. The instruments will function properly for about 7 years.. vertical stretch by e ; g(x) = e e x. A. E. D 6. C 7. F 8. B 9. B and G 0. never 1. always. never. sometimes a. (00) = 00 ( 1 + r ) () = ( 1 + r ) 0 ) ) ln = 0ln ( 1 + r ln 0 = ln ( 1 + r ln e 0 = 1 + r ln r = ( e 0-1) 0.11 A rate of about 1.1% will double the investment in years. b. (00) = 00 ( ) t = t ln = t ln t = ln ln It will take about 0 years to double the investment at a rate of.%. c. A() = 00 ( ) () Copyright by Holt, Rinehart and Winston. 97 Holt Algebra

24 The total amount will be about $ after years.. C 6. A 7. B 8a. vertical stretch by a factor of b. horizontal translation units right; D: {x x > } c. horizontal stretch by a factor of d. Possible answer: horizontal compression since 0.9 t = 0.97 kt log 0.9, where k = the constant log 0.97, which is > 1 9. Changing h translates the graph right (+) or left (-), and changing k translates the graph up (+) or down (-). 0. Possible answer: Translation left or right: Replace x with x - h. Translation up or down: Add k. Reflect across x-axis: Multiply b x by -1. Reflect across y-axis: Replace x with -x. Vertical stretch or compression: Multiply b x by a ±1. Horizontal stretch or compression: Divide x by c ±1. 1a. N(t) = (17)(0.99) t = 19(0.99 ) t m b. N(m) = 19(0.99 ) 1 c. m = 1 + = N(17) = 19(0.99 ) 1 1 There were about 1 soybean farms at the end of May TEST PREP, PAGE. A. H. B CHALLENGE AND EXTEND, PAGE a. translation 1 unit up b. horizontal compression by c. They are equivalent. d. log (x) = log + log x = 1 + log x 6. The value of y is undefined for x For (x - h) > 1, f(x) > 0. For (x - h) = 1, f(x) = 0. For 0 < (x - h) < 1, f(x) < 0. For (x - h) < 0, f(x) < 0 is undefined. SPIRAL REVIEW, PAGE 8. f(x) = x - x + = ( x - x) + = ( x - x) = (x - 1 ) + min: (1, ); D: ; R: {y y > }; 9. f(x) = x + x - = ( x + x ) - = ( x + = ( x + min: ( - x + 8) , ) ; D: ; R: {y y > }; 60. f(x) = - x - x + 1 6) - - ( = -( x + x) + 1 = - ( x + x + ) = - ( x + ) + max: ( -, ) ; D: ; R: {y y < }; 6) 61. f(x) 0.0 x x x ln e x + = x + 6. ln e -x = -x 6. e ln (x - 1) = x e ln x = x 7-8 CURVE FITTING WITH EXPONENTIAL AND LOGARITHMIC MODELS, PAGES -1 CHECK IT OUT! PAGES -7 1a. yes; 1.. no ed: s/b a??. B(t) 199(1. ) t ; ed: s/b a? (1. ) t t ln 000 t ln ln 000 t 199 ln 1.. The number of bacteria will reach 000 in about. min.. S(t) ln t; ln t ln t ln t 7.1 t e The speed will reach 8.0 m/s in about 16.6 min. Copyright by Holt, Rinehart and Winston. 98 Holt Algebra

25 THINK AND DISCUSS, PAGE 7 1. if there is a common ratio between the data values. Possible answer: There is only 1 ratio between data values, so there is no way to determine if that ratio is constant.. EXERCISES, PAGES 8-1 GUIDED PRACTICE, PAGE 8 1. exponential regression. no. yes;. no. yes; 6. T(t) 11(0.9 ) t ; 0 > 11(0.9 ) t 0 11 > 0.9 t ln 0 > t ln ln 0 t > 11 ln 0.9 > 1. It will take about 1. min for the tea to reach the temperature. 7. P(t) ln t; ln t ln t ln t 778. t e 11 1 It will take about 1 mo for the population to reach PRACTICE AND PROBLEM SOLVING, PAGES no 9. no. yes; yes; 1. P (t) a 9.(1.0 ) t ; < 1.6(1.0 ) t 1.6 < 1.0 t ln < t ln ln t > 1.6 ln 1.0 > = 07 The population will exceed million in T(t).(1.17 ) t ; 0 <.(1.17 ) t 0. < 1.17 t ln 0. < t ln1.17 ln 0 t >. ln 1.17 > = 011 The number of telecommuters will exceed 0 million in t(p) ln p; t(00) ln = 00 The population will reach 00 in year yes; f(x) = 1.(7. ) x 16. yes; f(x) = (0. ) x 17. Possible answer: a third data point because points can be fit by many different functions 18. r(d).99(0.999 ) d ; r(000).99(0.999 ) The calf survival rate is 1.0 per 0 cows at snow depths of 000 mm. 19. Use year 00 as the starting year (t = 0). s(t) 68.(.6878 ) t ; t = + = s() 68.(.6878 ) 6,,00 The sales will be about 6,,00 in three years. 0. Possible answer: an exponential decay model; the ratios are nearly constant. 1. Possible answer: nth differences have the same common ratio as first differences. a. F(t) = 011.6(0.98 ) t b = = 1.6% c. t = = 0 F(0) = 011.6(0.98 ) 0 There will be about 1,0,000 acres of farmland in 0. a. t = 1; 0 - S = 0(0.79 ) S = 79. S = 0. t = ; 0 - S = 0(0.79 ) 0 - S = 6.0 S = 6.8 t = S = 0(0.79 ) S = 16.0 S = 8.0 The speed is 0. mi/h at 1 s, 6.8 mi/h at s, and 8.0 mi/h at 8 s. b. s = 0 ( 0. 8 t ) ; the value of a is accurate to the nearest whole number; the value of b is accurate to the nearest thousandth.. Possible answer: There will be a constant ratio between consecutive values rather than constant differences. a. exponential b. Linear; the log of an exponential function of x is linear. TEST PREP, PAGE 0 6. C 7. F Copyright by Holt, Rinehart and Winston. 99 Holt Algebra

26 8.. (. ) = 6.1 CHALLENGE AND EXTEND, PAGE = a b Solve the system for a and b. 00 = a b Substitute for a: 00 = ( 8 b ) b. Solve for b: 6. = b b =. (b > 0). Solve for a: a = So f(x) = 7.68(. ) x. 0a. f(t) 0.01 ( t ) b. The initial concentration was 0.01 mg/c m, and it was not above the health risk level. c > 0.01 ( 0.96 t ) > 0.96 t ln > t ln ln t > ln 0.96 > 7.7 This will be about 7 hours later. 1. Exponential: {y y 0} causes an error. Logarithmic: {x x 0} or {y y 0} causes an error. SPIRAL REVIEW, PAGE 1. -x = -x = ± x = ±9. x - = x - = ± x = ± x = 7 or 6. x + x - = 0 (x - 1)(x + ) = 0 x - 1 = 0 or x + = 0 x = 1 or x = - 8. x + x + 1 = 0 ( x + x + 6) = 0 (x + )(x + ) = 0 x + = 0 or x + = 0 0. x = - or x = - 6 = x + -6 = ( ) x + -6 = (x + ) - = x + x = -8 x. 8 x = ( ) x + ( ) = ( -1 ) x + -(x + ) x = x = -x - x = - x = -1. x + = 0 x + = 0 x = -. x + 1 = x = 9 x = ±9 x = ± 9 7. x + x = 0 x(x + 8) = 0 x = 0 or x + 8 = 0 x = 0 or x = x + 9x - 6 = 0 (x - )(x + 1) = 0 x - = 0 or x + 1 = 0 x = or x = x = x + ( ) x = x + x = x + x = x =. 1 6 x = 6 x ( 6 ) x = 6 x x = x x = 0 MULTI-STEP TEST PREP, PAGE 1. t = = 0 A(0) = 17 e 0.0(0) = 17; t = = 0 A(0) = 17 e 0.0(0) = 19. The average farm size is 17 acres in 190 and 19. acres in = 17 e 0.0t 0 17 = e 0.0t ln 0 17 = 0.0t ln 0 t = = 196 The average farm size reached 0 acres in (17) = 17 e 0.0t = e 0.0t ln = 0.0t t = ln It took about 1. years for the average farm size to double.. N(t) = 6.1(0.98 ) t. t = = 70 N(70) = 6.1(0.98 ) There will be about 1.9 million farms in %(6.1) = 6.1(0.98 ) t 0. = 0.98 t ln 0. = t ln 0.98 t = ln 0. ln It takes about. years for the number of farms to decrease by 0% > 6.1(0.98 ) t 6.1 > t ln > t ln ln t > 6.1 ln 0.98 > = 00 The number of farms will fall below 1 million in 00. READY TO GO ON? PAGE 1. x = x. 9 x + < 7 7 x = - ( 7 ) x + x < 7 x = - (x + ) < x x < -8 x < - 16 Copyright by Holt, Rinehart and Winston. 00 Holt Algebra

27 . 1 x - 1 = 91 (x - 1)ln 1 = ln 91 x - 1 = ln 91 ln ln 91 x = ln 1 x 0.9. lo g (x - 1) x - 1 x 6 7. log 16x - log = log 16x = log x = x = x = 9. A = P(1 + r ) n A P = (1 + r ) n. 1.. x + = 0 (x + )ln = ln 0 x + = ln 0 ln x = ln 0 6. lo g x = x = ( x ) ln - x 0. = ( ) x = 1 =, log x + log (x + ) = 1 log x(x + ) = 1 x(x + ) = x + x - = 0 (x - )(x + ) = 0 x - = 0 or x + = 0 x = or x = - x = since x must be positive. log A = n log (1 + r) P log A n = P log (1 + r) log n ( ) log ( ) 0. It will take about 0. quarters, or.07 yrs, for the account to contain at least $ x 1. ln e = 1. ln e = x 16. e ln (1 - a) = 1- a 17. ln e b + = b + 18a. = e -70k ln = -70k ln k = b. N 00 = e (00) 8.87 There will be about 8.87 g left after 00 years. 1.; y = 0; The graph is stretched vertically by a factor of 1.. 0; x = -1; The graph is shifted 1 unit left and stretched vertically by a factor of.; D: {x x > - 1}. f(x) = -0. x. yes; constant ratio: ; f(x) = (1.) x 0.. 1; y = 0; The graph is stretched horizontally by a factor of ; x = -; The graph is shifted units left and reflected across the x-axis; D: {x x > -}. linear function; f(x) = 0.9x STUDY GUIDE: REVIEW, PAGES natural logarithmic function. asymptote. inverse relation LESSON 7-1, PAGE. growth 6. decay 8. growth 9. P(t) = 76(1.0 ) t. growth 7. growth Copyright by Holt, Rinehart and Winston. 01 Holt Algebra

28 yr LESSON 7-, PAGE P T = P L (1-0.0) 1. P L = P T K = 8 M; mi = 8 () km = 0 km LESSON 7-, PAGE 17. lo g = 18. lo g 9 1 = lo g 7 = - 0. = = = x y D: {x x > 0}; R: LESSON 7-, PAGE 6 9. lo g 8 + lo g 16 lo g (8 16) lo g log 0 + log,000 log (0,000) log 1,000, lo g 18 - lo g lo g 18 lo g 6 6. lo g lo g ( ) lo g. L = log I L = log I. log - log 0.1 log 0.1 log 0. log log 1 0 log ( ) 9 log 9 L = I L I = L + = log I L + = log I L + = I L + I = L + L L + - L = = The sound today was times more intense than the sound yesterday. LESSON 7-, PAGE 6 6. x - 1 = 7. 9 ( x - 1 = - x - 1 = - x = log x >..log x >. log x > 1 x > ) x 6 ( -1 ) x 6 -x 6 x = 0( ) n = 1.0 n ln = n ln 1.0 n = ln ln It will take about years to double the money. LESSON 7-6, PAGE 6 0a. t = = 6 19 = e 6k 19 = e 6k ln 19 = 6k ln 19 k = Copyright by Holt, Rinehart and Winston. 0 Holt Algebra

29 b. t = = 80 P(80) = e 0.06(80) 9 The population will be about 9 in 00. LESSON 7-7, PAGE 7 1. g(x) = - e x -. y = 0.6; y = 0; vertically compressed by a factor of and horizontally compressed by a factor of 6. V (t) = 00(1-0. ) t.. vertically stretched by a factor of 00 LESSON 7-8, PAGE 7 6. f(x) 11.6(1.0 ) x 7. f(x) ln x y = -0.; x = -0.; translated unit left and vertically stretched by a factor of 8. the exponential function; r 0.9 versus r 0.60 for the logarithmic function CHAPTER TEST, PAGE 8 1. decay. growth. decay. growth. f(x) = 100(1-0.1 ) t The value will fall below $000 in the th year. 6. f -1 (x) = x f -1 (x) = 6 (x ) 8. f -1 (x) = 6 ( x). lo g 16 = 1. ( 9. f -1 (x) = 6 ( x) 11. lo g 16 ) - = f -1 (x) = lo g x = -0. = 1. f -1 (x) = lo g. x D: {x x > 0}; R: D: {x x > 0}; R: 16. f -1 (x) = -lo g x D: {x x > 0}; R: 17. lo g 18 - lo g 8 lo g 18 8 lo g lo g lo g lo g (1.8 ) lo g 6 6 Copyright by Holt, Rinehart and Winston. 0 Holt Algebra

30 19. lo g lo g ( ) lo g x 1. x - 1 = 79 x x - 1 = ( 6 ) x - 1 = 6 ( ) x x - 1 = x x = -. lo g (x + 8) = x + 8 = x = 16. > 1(0.9 ) x > 0.9 x lo g x = x x 1. - x 1. - x x -0.. log 6 x - log x = 1 log ( 6 x x ) = 1 log x = 1 x = x = ln > x ln 0.9 ln x > ln 0.9 > 1. It will take about 1. min to reduce to less than ml. = e -000k ln = -000k ln k = e () There will be about g left after years. 7. f(x) = ln (x + ) f(x) = ln x; 0 < ln x 1.6 <.6 ln x < ln x 1.6 x > e.6 > 8.1 The population will exceed 0 in year E; y = lo g ( x - = -x x = 88 x = 8 (8) - ) = lo g = - lo g 9 {lo g [lo g (x)]} = lo g [lo g (x)] = 9 lo g (x) = x = 8 = 6,6 COLLEGE ENTRANCE EXAM PRACTICE, PAGE 9 1. A. D. C; e x e. = e.x e x +. = e.x x +. =.x x =. A; lo g ( x - ) = lo g (. B; lo g 7 9 lo g 9 lo g 7 -x + 6 ) Copyright by Holt, Rinehart and Winston. 0 Holt Algebra