C H A P T E R Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs........ 7 Section. Properties of Logarithms................. 8 Section. Eponential and Logarithmic Equations......... 89 Section. Eponential and Logarithmic Models.......... Review Eercises............................ 7 Problem Solving............................. 9 Practice Test..............................
C H A P T E R Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs You should know that a function of the form f a, where a >, a, is called an eponential function with base a. You should be able to graph eponential functions. You should know formulas for compound interest. For n compoundings per ear: A P r n nt. For continuous compoundings: A Pe rt. Vocabular Check. algebraic. transcendental. natural eponential; natural. A P. A Pe n r nt rt. f... 9.8. f...88. f.. f... g.. 77.77 f...7 7. f 8. f rises to the right. 9. Increasing Asmptote: Asmptote: Intercept:, Intercept:, Matches graph (c). Matches graph (d). f Decreasing Asmptote: Intercept:, Matches graph.. f rises to the right.. f Asmptote: Intercept:, Matches graph. f.. Asmptote:
Chapter Eponential and Logarithmic Functions. f. f f.. f.7.8 Asmptote: Asmptote:. f. f f.8.7 f... Asmptote: Asmptote:. f f.... 7 Asmptote: 7. f, g 8. Because g f, the graph of g can be obtained b shifting the graph of f four units to the right. f, g Because g f, the graph of g can be obtained b shifting the graph of f one unit upward. 9. f, g. Because g f, the graph of g can be obtained b shifting the graph of f five units upward. f, g Because g f, the graph of g can be obtained b reflecting the graph of f in the -ais and shifting f three units to the right. (Note: This is equivalent to shifting f three units to the left and then reflecting the graph in the -ais.)
Section. Eponential Functions and Their Graphs 7. f 7, g 7. Because g f, the graph of g can be obtained b reflecting the graph of f in the -ais and -ais and shifting f si units to the right. (Note: This is equivalent to shifting f si units to the left and then reflecting the graph in the -ais and -ais.) f., g. g f, hence the graph of g can be obtained b reflecting the graph of f in the -ais and shifting the resulting graph five units upward.... f. 7. f e.7 8. f e e.. 9. f e.87. f.e. f e. 7.7..e.9 f e. e. 79.7. f e. f e f..8.78 7.89 f 7.89.78.8. Asmptote: Asmptote:. f e. f e. 8 7 f..9.. f.7.97..7 Asmptote: Asmptote: 8 7 8 7
8 Chapter Eponential and Logarithmic Functions 7. f e 8. f e f.7..7.7 f.7..8.78 Asmptote: Asmptote: 9 8 7 7 8 7 7 8 9..8..8. st e.t 7 7 8. st e.t. g e. h e 7. 7. 7. 7 8. 9. e e. e e. e e. or e e or
Section. Eponential Functions and Their Graphs 9. P $, r.%, t ears Compounded n times per ear: A P n r nt. n n Compounded continuousl: A Pe rt e. n Continuous Compounding A $. $.9 $7.7 $9. $. $.. P $, r %, t ears Compounded n times per ear: A. n n Compounded continuousl: A e. n Continuous Compounding A $8. $8.9 $88.8 $9.8 $9.79 $9.8. P $, r %, t ears Compounded n times per ear: A P r n nt. n n Compounded continuousl: A Pe rt e. n Continuous Compounding A $.8 $. $. $.89 $.8 $.. P $, r %, t ears Compounded n times per ear: A. n n Compounded continuousl: A e. n Continuous Compounding A $,8.7 $,.89 $,88. $,97. $,. $,.8 7. A Pe rt,e.t t A $7,9.9 $,7.9 $9,8. $9,.9 $88,8.7 8. A Pe rt,e.t t A $,8. $9,8. $7,9.77 $,78. $,.
7 Chapter Eponential and Logarithmic Functions 9. A Pe rt,e.t t A $,98.9 $,. $8,. $,.8 $9,8.8. A Pe rt,e.t t A $7,8.8 $,. $,9.8 $8,. $9,.. A,e.87. $,8.7 A e.7. C.9. $. $,.. p e.. Vt e.t V,.98 computers V.,.7 computers (c) V,,9. computers When : p e. $. (c) Since,. is on the graph in part, it appears that the greatest price that will still ield a demand of at least units is about $.. P.e.9t 7. Since the growth rate is negative, the population is decreasing..9.9%, In 998, t 8 and the population is given b P8.e.98 7.8 million. In, t and the population is given b P.e.9. million. (c) In, t and the population is given b P.e.9.8 million. Q t99 (c) Q grams Q. grams 8. Q t7 When t : Q 7 (c) Q grams When t : Q 7 7.8 grams Mass of C (in grams) 8 8 Time (in ears) t
Section. Eponential Functions and Their Graphs 7 9. 7e.9 Sample Data Model.. 8 8.8 (c) When : (d).%. 7e.9 7e.9 when 7 9 9.9 8 masses. 99 99. 7. P p 7,8e.h Atmospheric pressure (in pascals),, 8,,,, 7,8e.8,7 pascals Altitude (in km) h 7. True. The line is a horizontal asmptote for the 7. False, e 7,8 99,99. e is an irrational number. graph of f. 7. f 7. Thus, 9 h f g, but f h. g h Thus, g h but g f. 7. f and f 7. f g Thus, f g h. h g h Thus, none are equal.
7 Chapter Eponential and Logarithmic Functions 77. and = = 9 9 < when <. > when >. 78. f e g 7 Decreasing:,,, Increasing:, Relative maimum:, e Relative minimum:, Decreasing:., Increasing:,. Relative maimum:.,. 79. f. (Horizontal line) and g e. 8. The functions (c) and (d) are eponential. f g As, f g. As, f g. 8. 8. ± and, 8. f 9 Vertical asmptote: 9 9 Horizontal asmptote: 8 7 8 f 9
Section. Logarithmic Functions and Their Graphs 7 8. f 7 8. Answers will var. Domain:, 7 9 7 8 Section. Logarithmic Functions and Their Graphs You should know that a function of the form log a, where a >, a, and >, is called a logarithm of to base a. You should be able to convert from logarithmic form to eponential form and vice versa. log a a You should know the following properties of logarithms. log a since a. log a a since a a. (c) log a a since a a. (d) a log a Inverse Propert (e) If log a log a, then. You should know the definition of the natural logarithmic function. log e ln, > You should know the properties of the natural logarithmic function. ln since e. ln e since e e. (c) ln e since e e. (d) e ln Inverse Propert (e) If ln ln, then. You should be able to graph logarithmic functions. Vocabular Check. logarithmic.. natural; e. a log a.. log. log 8 8. log 7 9 7 9. log. log. log 8 8 7. log 8. log 8 8 9. log. 8 log 8. 8 log 8. 9 7 log 9 7
7 Chapter Eponential and Logarithmic Functions. log. log. 7 log 7.. log. 7. f log 8. f log since f log f log since 9. f log 7. f log. g log a f log 7 since 7 f log since ga log a a b the Inverse Propert. g log b. gb log b b since b b f log. f log.97 f f log log.99. f log. f..97 f log 7. log since f 7..877 8. log. 9. since. Since., log.. log. 9 log 9 Since a log a, 9 log 9.. f log. g log Domain: > The domain is,. Domain:, -intercept:, -intercept:, Vertical asmptote: log Vertical asmptote: log f. log. h log Domain:, Domain: > > -intercept: log log 9 8 The domain is,. -intercept: log 8 The -intercept is 9,. Vertical asmptote: The -intercept is,. log Vertical asmptote: log log 7 9 7 9
Section. Logarithmic Functions and Their Graphs 7. f log. log Domain: > > Domain: > > The domain is,. -intercept: log log The -intercept is,. Vertical asmptote: log log The domain is,. -intercept: log log The -intercept is,. Vertical asmptote: log...8.. f 7. log 7 Domain: The domain is,. -intercept: log > > 8 The -intercept is,. Vertical asmptote: The vertical asmptote is the -ais..7....8. 8. log Domain: > < The domain is,. -intercept: log The -intercept is,. Vertical asmptote: log
7 Chapter Eponential and Logarithmic Functions 9. f log. Asmptote: Point on graph:, Matches graph (c). The graph of f is obtained b shifting the graph of g upward two units. f log Asmptote: Point on graph: Matches graph (f). f, reflects g in the -ais.. f log. Asmptote: Point on graph:, Matches graph (d). The graph of f is obtained b reflecting the graph of g about the -ais and shifting the graph two units to the left. f log Asmptote: Point on graph:, Matches graph (e). f shifts g one unit to the right.. f log log. Asmptote: Point on graph:, Matches graph. The graph of f is obtained b reflecting the graph of g about the -ais and shifting the graph one unit to the right. f log Asmptote: Point on graph:, Matches graph. f reflects g in the -ais then reflects that graph in the -ais.. ln.9... e.9.... ln.9... e.9... 7. ln.8... e.8... 8. ln.... e.... 9. ln.... e..... ln 79.... e.... 79. ln e. ln e e e. e.8... ln.8.... e 7.89... ln 7.89.... e.87... ln.87.... e.9... ln.9... 7. e..... ln..... 8. e..... ln..... 9. e ln. e ln. f ln. f8. ln 8..9 f ln f. ln..8. g ln. g.7 ln.7.7 g ln g ln.9
Section. Logarithmic Functions and Their Graphs 77. g ln. ge ln e b the Inverse Propert g ln ge ln e 7. g ln 8. ge ln e b the Inverse Propert g ln ge ln e 9. f ln 7. h ln Domain: > > Domain: > > The domain is,. The domain is,. -intercept: ln e The -intercept is,. -intercept: ln e 8 Vertical asmptote: The -intercept is,. f..9.9. Vertical asmptote: ln e.9.7.9 9.9 7. g ln 7. f ln Domain: > < Domain: > < The domain is,. The domain is,. -intercept: ln -intercept: ln e The -intercept is,. e Vertical asmptote: The -intercept is,. g..9.9. Vertical asmptote: ln e.9.8..8 7. log 7. f log 7. ln 9
78 Chapter Eponential and Logarithmic Functions 7. f ln 77. ln 78. f ln 9 79. log log 8. log log 9 9 8. log log 8. 7 log log 9 9 8. ln ln 8. ln ln 8. ln ln 8. ± ln ln or 87. t. ln, > When $.: t. ln. ears. When $.8: t. ln.8 ears.8 Total amounts:. $9,. (c) Interest charges:.8 $,. 9,, $,,., $,. (d) The vertical asmptote is. The closer the pament is to $ per month, the longer the length of the mortgage will be. Also, the monthl pament must be greater than $. 88. t ln K.9 K 8 t 7.. 8.9.9.. The number of ears required to multipl the original investment b K increases with K. However, the larger the value of K, the fewer the ears required to increase the value of the investment b an additional multiple of the original investment. t 8 K
Section. Logarithmic Functions and Their Graphs 79 89. ft 8 7 logt, t 9. log I log log decibels (c) f 8 7 log 8. f 8 7 log 8. log log decibels (c) No, the difference is due to the logarithmic relationship between intensit and number of decibels. (d) f 8 7 log. 9. False. Reflecting g about the line will determine 9. True, log 7 7. the graph of f. 9. f, g log 9. f, g log 9. f e, g ln f g f g f g f and g are inverses. Their graphs are reflected about the line. f and g are inverses. Their graphs are reflected about the line. f and g are inverses. Their graphs are reflected about the line. 9. f, g log 97. f ln, g f g f and g are inverses. Their graphs are reflected about the line. The natural log function grows at a slower rate than the square root function. f ln, g The natural log function grows at a slower rate than the fourth root function. g f g f, 98. f ln f....9.8 As, f. (c).
8 Chapter Eponential and Logarithmic Functions 99. False. If were an eponential function of, then a, but a a, not. Because one point is,, is not an eponential function of. (c) True. a For a,.,,, 8 True. log a For a, log., log, log 8, log 8 (d) False. If were a linear function of, the slope between, and, and the slope between, and 8, would be the same. However, m and m 8. Therefore, is not a linear function of.. log so, for eample, if there is no value of for which a a, a,. If a, then ever power of a is equal to, so could onl be. So, log a is defined onl for < a < and a >.. f ln 8 Increasing on, Decreasing on, (c) Relative minimum:,. h ln 8 9 9 Increasing on, Decreasing on, (c) Relative minimum:, For Eercises 8, use f and g.. f g f g. 8 7 f g Therefore, f g.. fg f g. f g Therefore, g f. 7. f g7 f g7 8. f 7 f 8 g f ( g f g Therefore, g f 7.
Section. Properties of Logarithms 8 Section. Properties of Logarithms You should know the following properties of logarithms. log log a ln a log log a log b log b a log a ln a log a uv log a u log a v (c) log a u v log a u log a v (d) log a u n n log a u lnuv ln u ln v u ln ln u ln v v ln u n n ln u You should be able to rewrite logarithmic epressions using these properties. Vocabular Check. change-of-base. log ln log a ln a. log a uv log a u log a v. ln u n n ln u This is the Product Propert. Matches (c). This is the Power Propert. Matches.. log a u v log a u log a v This is the Quotient Propert. Matches.. log log log log ln ln. log log log log ln ln. log log log log ln ln. log log log log ln ln. log log log log ln ln. log log log log ln ln 7. log. log log. log. ln ln. 8. log 7. log log 7. log 7. ln ln 7. 9. log 7 log 7 ln 7.77 log ln. log 7 log ln log 7 ln 7.7. log log ln. log ln. log log ln log ln.. log log. ln. 9..7 log 9 ln 9. log. log. log ln. ln.9. log log ln. log ln. log. log. log ln. ln.8 7. log 8 log 8 log log log
8 Chapter Eponential and Logarithmic Functions 8. log log log 9. log log log log log log log log log. log log log log log log 9 log log log log log log log log. lne ln ln e. ln ln ln e ln ln e. log 9 log ln ln e ln. log log log. log 8 log log. log log log 7. log..log. log.. 8. log 8.. log 8 9. log 9 is undefined. 9 is not in the domain of log.. log..8. log is undefined because. ln e... is not in the domain of log. ln e ln e. ln ln lne. e ln e ln e ln e. ln e ln e 7 ln e. ln e ln e ln e ln e 7. ln e e ln e 7 7 log 7 log log 7 log log log 8. log log log log log log 9. log log log
Section. Properties of Logarithms 8. log z log log z. log 8 log 8. log log log. log log log. log z log z. lnz ln z ln z log. ln t ln t ln t 7. ln z ln ln ln z 8. ln ln ln z log log log log log log log 9. ln zz ln z lnz. ln z lnz, z > ln ln ln ln ln ln ln ln. a log log 9 a log 9. ln ln ln log a log ln ln log a log, a > ln ln. ln ln. ln ln ln ln ln ln ln ln ln ln ln ln ln. ln z ln ln z. ln ln ln z log log z log z log log log z ln ln ln z log log log z 7. log z log log z 8. log log log z log log log z log z log log z log log log z log log log z 9. ln ln. ln ln ln ln ln ln ln ln ln ln ln ln ln
8 Chapter Eponential and Logarithmic Functions. ln ln ln. ln ln t ln t ln t. log z log log z. log 8 log t log 8 t. log log. log 7z log 7 z 7. log log log 8. log log log 9. ln ln ln ln 7. ln ln 8 lnz ln 8 lnz ln lnz ln z 7. log log log z log log log z 7. log log z log z log log log z log log log z log log z log z 7. ln ln ln ln ln ln ln ln ln ln 7. ln z lnz lnz ln zz lnz lnzz lnz ln z z z 7. ln ln ln ln ln ln ln ln ln ln 7. ln ln ln ln ln ln ln ln ln ln ln ln ln
Section. Properties of Logarithms 8 77. log 8 log 8 log 8 log 8 log 8 log 8 log 8 log 8 log 8 log 8 log 8 78. log log log log log log log log log log log 79. log log log log log The second and third epressions are equal b Propert. 8. log 7 7 log 7 7 log 7 7 log 7 8. log 7 log 7 log 7 b Propert and Propert log I log I log log I log I When I : log decibels 8. log I 8. Difference log. log log. 7 log. log. 7. log.79 log.79 db. 7 logi log log I log I log With both stereos plaing, the music is decibels louder. log
8 Chapter Eponential and Logarithmic Functions 8. f t 9 logt, t f t 9 logt f 9 (c) f 9 log 79. (d) f 9 log 7. (e) 9 (f) The average score will be 7 when t 9 months. See graph in (e). (g) 7 9 logt logt logt t t 9 months 7 8. B using the regression feature on a graphing calculator we obtain..8 ln. 8. 8 (c) t (in minutes) T C T C lnt T (d) T..9 t See graph in..t. T.7 T..9 t T.t. 8 78 7..7.87. 7...97.7...8.....9...8. 9. 8..9.8 (e) Since the scatter plot of the original data is so nicel eponential, there is no need to do the transformations unless one desires to deal with smaller numbers. The transformations did not make the problem simpler. Taking logs of temperatures led to a linear scatter plot because the log function increases ver slowl as the -values increase. Taking the reciprocals of the temperatures led to a linear scatter plot because of the asmptotic nature of the reciprocal function. lnt.7t T e.7t This graph is identical to T in. 87. f ln 88. fa fa f, a >, > False, f since is not in the domain of f. True, because fa ln a ln a ln fa f. f ln
Section. Properties of Logarithms 87 89. False. f f ln ln ln ln 9. f false f; f ln can t be simplified further. f ln ln ln f 9. False. 9. If f <, then < <. fu fv ln u ln v ln u ln v u v True 9. Let log and log then b u and b b u b v, v. u v b b b Then log b uv log b b log b u log b v. 9. Let log then u b and u n b n b u,. log b u n log b b n n n log b u 9. f log log ln log ln 9. f log log ln log ln 97. f log log log ln ln 98. f log 99. f log.8. f log. log log ln ln log ln log.8 ln.8 log log. ln ln.. f ln ln, g, h ln ln ln f h b Propert g f = h
88 Chapter Eponential and Logarithmic Functions. ln.9, ln.98, ln.9 ln.9 ln.98 ln ln ln ln.9.9.8 ln.9 ln ln ln ln.9.98.797 ln 8 ln ln.9.79 ln 9 ln ln.98.97 ln ln ln ln.9.9. ln ln ln ln ln ln.9.98.88 ln ln ln ln.9.98.78 ln ln ln.9.77 ln 8 ln ln ln ln ln.98.9.89 ln ln ln ln ln ln.9.9.99.,. 7 8. 8 8 8 8 if,.. 7. 8. The zeros are,. 9.. 8 ± 8 ± 97 ± ± ± The zeros are ±.
Section. Eponential and Logarithmic Equations 89 Section. Eponential and Logarithmic Equations To solve an eponential equation, isolate the eponential epression, then take the logarithm of both sides. Then solve for the variable.. log. ln e a a To solve a logarithmic equation, rewrite it in eponential form. Then solve for the variable.. a log a. e ln If a > and a we have the following:. log a log a. a a Check for etraneous solutions. Vocabular Check. solve.. etraneous (c) (d). 7. 7 Yes, is a solution. 7 No, is not a solution. No, is not a solution. 7 8 No, is not a solution.. e 7. (c) e e e e 7 No, e is not a solution. e ln e ln 7 Yes, ln is a solution. e.9 e.9 7 Yes, e ln.9.9 is a solution. e (c) e ln eln Yes, No, ln ln ln ln ln ln e.8 is a solution. e [ln ln ln ln e 97.999 9.999 is not a solution.. e ln e. e.79. Yes,. is an approimate solution.
9 Chapter Eponential and Logarithmic Functions. log. (c). Yes,. is an approimate solution. No, is not a solution.. Yes, is a solution. log log log Since, is a solution. 7 log 7 log Since, 7 is not a solution. (c) 97 log 97 log Since, is not a solution. 7. ln.8 8. ln.8 ln ln.8 lnln.8.8 No, ln.8 is not a solution. e.8 ln e.8 lne.8.8 Yes, e.8 is a solution. (c). ln. ln..8 Yes,. is an approimate solution. ln.8 (c) Yes, e.8 is a solution. ln.7 ln.7.8 Yes,.7 is an approimate solution. ln ln.8 lnln.8.89 No, e.8 ln e.8 ln e.8.8.7 ln.8 ln.8 is not a solution. 9..... ln ln. ln ln. e. e ln ln ln ln ln e ln ln e ln ln ln.9.8 7. ln 8. ln 7 9. log. log e ln e e ln e 7 log e e 7 or.8.8.9
Section. Eponential and Logarithmic Equations 9. f g. 8 Point of intersection:, 8 f g. 7 9 7 7 Point of intersection:, 9 f g. log 9 Point of intersection: 9, f g ln e ln e Point of intersection:,. e e. e e 8 7. e e 8 8 or, B the Quadratic Formula.8 or.8. 8. e e 9.. log log log, log log log. or ln ln ln ln.7. e. e 9. e 9 9 e e 9 e 8 ln e ln ln e ln 9 ln e ln 8 ln.9 ln 9. ln 8.. 7. 8. 7 log 7 ln 7 ln. ln ln 8 ln ln 8 ln 8.99 ln ln ln ln ln ln ln ln ln.89 7. t. 8. t. 9. 7 t ln t ln. t ln ln. t t t t ln. ln ln. t. ln. log 8
9 Chapter Eponential and Logarithmic Functions.. ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln. 8 ln 8 ln ln 8 ln ln 8 ln 8 ln ln 8 ln ln 8 ln 8 ln ln ln ln ln 8.97. 8. 7. 8 log log log log 7 log log 7 log 7 log 7 7 ln ln 7 ln ln 7 ln 7 ln ln 7.9 ln.9.. 8 7. e 8. e ln ln e ln ln ln ln ln ln ln ln.88 ln ln.9 ln ln ln. ln 9. e. e 7. 7 e. e e e e e ln ln e ln e e ln ln ln ln e ln ln. ln ln.8.
Section. Eponential and Logarithmic Equations 9. 7 9. 8 8 8 8 log log 8 log 8 log8 log log8 log or ln8 ln.8. log. ln. ln ln. ln ln..8 ln. e e. e e e or e (No solution) ln.9 e e e e e or e ln.9 or ln.99 7. e e 8. e e e e Not possible since e > for all. e e ln.8 e 9e e 9e Because the discriminant is 9, there is no solution. 9. e. e e e 7 ln 7 ln 7 8.. e e 8 e 7 8 e 7 e 7 ln ln e 7 e e e 98 e ln 98 ln 98. ln 7 ln 7 ln 7 ln 7.9
9 Chapter Eponential and Logarithmic Functions. 9 e 7 9 7e 7 e e ln ln e ln ln.7.. t ln. t ln t ln. ln t ln ln....7 9t..98 9t ln.98 9t ln 9t ln.98 ln t ln.7 9 ln.98. t ln. t ln t ln. ln t ln ln..9..878 t 7. g e Algebraicall: ln.878 t ln t ln.878 ln t ln ln.878.9 e e ln ln.7 The zero is.7. 8. f e 9. f e 9 e e.7 e ln.7 ln.7 ln.7 Algebraicall: e 9 e 9 ln 9 9. The zero is.. ln 9.87 The zero is.87.
Section. Eponential and Logarithmic Equations 9 7. g 8e 7. 8e 7 e.7 ln.7. ln.7.78 The zero is.78. gt e.9t Algebraicall: e.9t.9t ln t ln.9 t.7 The zero is t.7. 8 7. f e.8 7 7. ht e.t 8 7. f e.7 9 e.8 7 Algebraicall: e.7 9 e.8 7 e.t 8.7 ln 9 e.8 7.8 ln 7 ln 7.8 e.t 8.t ln 8 t ln 8. ln 9.7. The zero is...8 t. The zero is.8. The zero is t.. 7 7. ln 7. ln 77. ln. 78. ln e. e ln e e. e ln e e 7.89 e.. e e.8 79. log 8. log z 8. ln 8. ln 7,,. log z z z. ln e e. ln 7 e ln e 7 e 7. 8. ln 8. ln 8 8. 7 ln 8. ln e e ln8 e ln ln 8 e 8 e ln ln e 8 e e e ln e.89 e 8. e,..
9 Chapter Eponential and Logarithmic Functions 87. log. 88. log. log...988 log log log.89 89. ln ln 9. ln e e e e e e e e e e.7 This negative value is etraneous. The equation has no solution. ln ln ln e ln e e e The onl solution is ± e e.. 9. ln ln 9. ln e e ± e ± e ± e The negative value is etraneous. The onl solution is e.98. ln ln ln e ln e e e ± 9 e 9 e The onl solution is.79. 9. ln ln ln 9. ln ln or Both of these solutions are etraneous, so the equation has no solution. ln ln ln ln ln ± ±. (The negative apparent solution is etraneous.) 9. log log 7
Section. Eponential and Logarithmic Equations 97 9. log log 7 The apparent solution 7 is etraneous, because the domain of the logarithm function is positive numbers, and 7 and 7 are negative. There is no solution. 97. log log log 98. Quadratic Formula Choosing the positive value of (the negative value is etraneous), we have log log 7.. ± 7 log log log log log or The value is etraneous. The onl solution is. 99. log log. log log 8 log log log 8 log 8 8 9 8 9 9 9 or The value is etraneous. The onl solution is 9.. log 8 log 8 log The onl solution is 8 8 7 7 ± 7.8 (etraneous) or 8.8 9 8 8.8. 7 ±, 8 9 ± 8
98 Chapter Eponential and Logarithmic Functions. log log log log( 9, 9, The onl solution is ± 9, ±,, ± 7 78. etraneous or. 7... 7. From the graph we have.87 when 7. Algebraicall: 7 ln ln 7 ln ln 7 ln 7.87 ln 8 e e ln ln.97 The solution is.97. 8.. ln 8 ln ln From the graph we have.8 when. Algebraicall: ln ln e.8 ln. e ln e. e. e..8 The solution is.8.
Section. Eponential and Logarithmic Equations 99 7. A Pe rt A Pe rt 8. r. r. e.8t 7 e.8t A Pe rt A Pe rt e.8t e.8t e.t 7 e.t ln.8t ln.8 t ln.8t ln.8 t e.t ln ln e.t ln.t e.t ln ln e.t ln.t t 8. ears t.9 ears ln. t ln. t t.8 ears t 9. ears 9. p.e. p p.e..e. e. e.. ln. ln units 98 units. p. e When When p $:..88 e...88e..8.88e. e. ln ln e. ln. ln. e. e. units p $:.8.9 e..8.9e...9e. 8 e. ln 8 ln e. ln 8. ln8. e. e. 8 units. V.7e 8.t, t As t, V.7. (c)..7e 8.t Horizontal asmptote: V.7 The ield will approach.7 million cubic feet per acre.. e8.t.7 8. ln 7 t t 8. 9. ears ln7
Chapter Eponential and Logarithmic Functions. N 8.. When N : 8.. 8 log. 8 log 8..7 inches 7. ln t, t 7. ln t 8. ln t ln t. t e. t corresponds to the ear.. 8 88. ln t, t 9 8 88. ln t 9 88. ln t ln t 9. 88. t e.. Since t represents 99, t. indicates that the number of dail fee golf facilities in the U.S. reached 9 in.. From the graph shown in the tetbook, we see horizontal asmptotes at and. These represent the lower and upper percent bounds; the range falls between % and %. Males e.9.7 e.9.7. 9.7 ln. 9.7 e.9.7 9.7 inches Females e.7. e.7..7. ln.7. e.7.. inches. P.8 e.n. Horizontal asmptotes: P, P.8 The upper asmptote, P.8, indicates that the proportion of correct responses will approach.8 as the number of trials increases. (c) When P % or P.:..8 e.n e.n.8. e.n.8. ln e.n ln.8..n ln.8..8 ln. n trials.
Section. Eponential and Logarithmic Equations 7...88 ln.9....8.. 78....9 The model seems to fit the data well. (c) When :...88 ln.9 Add the graph of to the graph in part and estimate the point of intersection of the two graphs. We find that. meters. (d) No, it is probabl not practical to lower the number of gs eperienced during impact to less than because the required distance traveled at is.7 meters. It is probabl not practical to design a car allowing a passenger to move forward.7 meters (or 7. feet) during an impact. 8. T 7 h From the graph in the tetbook we see a horizontal asmptote at T. This represents the room temperature. 7 h 7 h 7 h h 7 ln ln h 7 ln h ln 7 ln7 ln h h.8 hour 9. log a uv log a u log a v. True b Propert in Section.. log a u v log a ulog a v False.. log log log. log a u v log a u log a v. False..9 log log a u v log a u log a v True b Propert in Section... Yes, a logarithmic equation can have more than one etraneous solution. See Eercise 9. log log. A Pe rt (c) A Pe rt Pe rt A Pe rt Pe rt e rt e rt Pe rt A Pe rt Pe rt e rt e rt Pe rt This doubles our mone. Doubling the interest rate ields the same result as doubling the number of ears. If > e rt (i.e., rt < ln ), then doubling our investment would ield the most mone. If rt > ln, then doubling either the interest rate or the number of ears would ield more mone.. Yes. Time to Double P Pe rt ln rt ln r t Time to Quadruple P Pert e rt e rt ln rt ln r t Thus, the time to quadruple is twice as long as the time to double.
Chapter Eponential and Logarithmic Functions. When solving an eponential equation, rewrite the original equation in a form that allows ou to use the One-to-One Propert a a if and onl if or rewrite the original equation in logarithmic form and use the Inverse Propert log a a. When solving a logarithmic equation, rewrite the original equation in a form that allows ou to use the One-to-One Propert log a log a if and onl if or rewrite the original equation in eponential form and use the Inverse Propert a log a. 7. 8 8. 9. 7.. f 9 Domain: all real numbers -intercept:, 9 -ais smmetr ± ± ± 9 8 8 8. 8 8. g,, Domain: all real numbers -intercept:, -intercept:, <... log 9 log 9 ln 9 log ln.. log log ln. log ln 7. log log ln log ln.9 8. log 8 log ln.8 log 8 ln 8
Section. Eponential and Logarithmic Models Section. Eponential and Logarithmic Models You should be able to solve growth and deca problems. Eponential growth if b > and ae b. Eponential deca if b > and ae b. You should be able to use the Gaussian model ae b c. You should be able to use the logistic growth model a be. r You should be able to use the logarithmic models a b ln, a b log. Vocabular Check. ae b ; ae b. a b ln ; a b log. normall distributed. bell; average value. sigmoidal. e. This is an eponential growth model. Matches graph (c). e. This is an eponential deca model. Matches graph (e). log This is a logarithmic function shifted up si units and left two units. Matches graph.. e. This is a Gaussian model. Matches graph. ln. This is a logarithmic model shifted left one unit. Matches graph (d). e This is a logistic growth model. Matches graph (f). 7. Since A e.t, the time to double is given b 8. Since A 7e.t, the time to double is given b e.t and we have 7e.t, and we have e.t ln ln e.t ln.t t ln 9.8 ears.. Amount after ears: A e. $9.7 7e.t e.t ln ln e.t ln.t t ln. ears.. Amount after ears: A 7e. $.
Chapter Eponential and Logarithmic Functions 9. Since A 7e rt and A when t 7.7, we have. Since A,e rt and A, when t, the following. we have 7e 7.7r e 7.7r ln ln e 7.7r ln 7.7r r ln.898 8.98% 7.7 Amount after ears: A 7e.898 $8.7,,e r e r ln ln e r ln r r ln Amount after ears:.77.77%. A,e.77 $7,87.97. Since A e rt and A $. when. Since A e rt and A 9, when t, we have t, we have the following. 9, e r. e r 9, er..% r ln. The time to double is given b e.t t ln. ears.. ln 9, ln er ln 9, r The time to double is given b e.t r ln9, t ln ears... or.%.. Since A Pe.t and A,. when t,. Since A Pe.t and A when t, we have we have the following. Pe.,. Pe. P,. e. $7.. P $7.8 e. The time to double is given b t ln The time to double is given b t ln.7 ears... ears..., P.7 P,.7, $,87.9.. A P r n nt, P. () P $.
Section. Eponential and Logarithmic Models 7. P, r % (c) n. t t ln. ln t ln. ears ln. n. t t ln. ln t ln ln.. ears. t t ln. ln (d) Compounded continuousl e.t.t ln n t t ln. ears. ln ln.. ears 8. P, r.%. n (c) t n t ln.9 ears ln. ln ln.. ears n t ln ln.. ears (d) Compounded continuousl t ln. ears. 9. P Pe rt r % % % 8% % % ln rt ln r e rt t t ln r (ears).9 7.7 8..7.99 9... Using the power regression feature of a graphing utilit, t.99r.. P P r t r % % % 8% % % ln ln r t ln t ln r ln ln r t r t t ln ln r (ears).8 8. 8.8.7. 9.9
[[ Chapter Eponential and Logarithmic Functions.. Continuous compounding results in faster growth. A.7t and A e.7t A. Using the power regression feature of a graphing utilit, t.r. Amount (in dollars)..7... A = e.7t A = +.7[[ t 8 Time (in ears) t. ( ) [[ t A = +. A = +.[[ t From the graph, % compounded dail grows faster than % simple interest. [[ [[. C Cek99.. e k99 ln. ln e k99 ln. k99 k ln. 99 Given C grams after ears, we have eln.99.8 grams. C Cek99 ek99 ln ln ek99 ln k99 k ln 99 Given. grams after ears, we have. Ce ln99 C. grams. 7. C Cek7 8.. e k7 ln. ln e k7 ln. k7 k ln. 7 Given grams after ears, we have Celn.7 C. grams. C Cek7 9. ek7 ln ln ek7 ln k7 k ln 7 Given C grams, after ears we have C Cek,. e k, ln. ln e k, ln. k, k ln., Given. grams after ears, we have. Celn., C. grams. e ln7. grams.
Section. Eponential and Logarithmic Models 7. C Cek,. ek, ln ln ek, ln k, k ln, Given. grams after ears, we have. Ce ln, C. grams. ln b ln ae b a e b Thus, e.77. ae b. b b.77 ln ln e b ln b ln ae b aeb a eb e b b b.7 Thus, e.7.. ln b ln ae b. ae b a e b eb b Thus, e.. b. ln ln eb ln ln ae b ae b a eb b b Thus, e.. b.. P e.9t Since the eponent is negative, this is an eponential deca model. The population is decreasing. For, let t : For, let t : P thousand people P 8.9 thousand people (c). million thousand e.9t e.9t ln.9t t ln.9 8.9 The population will reach. million (according to the model) during the later part of the ear 8.. Countr Bulgaria 7.8 7. Canada.. China 8.9 7. United Kingdom 9.. United States 8. 9. CONTINUED
8 Chapter Eponential and Logarithmic Functions. CONTINUED Bulgaria: For, use t. 7.8e.9.88 million China: a 7.8 7. 7.8e b ln 7. b b.9 7.8 a 8.9 7. 8.9e b ln 7. b b. 8.9 For, use t. 8.9e.. million United Kingdom: a 9.. 9.e b ln. b b.8 9. For, use t. 9.e.8.7 million Canada:..e b ln. b b.9. For, use t..e.9. million United States: a. a 8. 9. 8.e b ln 9. b b.9 8. For, use t. 8.e.9 7.9 million The constant b determines the growth rates. The greater the rate of growth, the greater the value of b. (c) The constant b determines whether the population is increasing b > or decreasing b <. 7. 8e kt 8. When t,,:, 8e k, 8 ln, 8 k When t : ek k ln,8.988 8e.988,9,7 hits e kt e k ln k k.7 For, t : e.7 $.98 million
Section. Eponential and Logarithmic Models 9 9. N e kt. e k e k ln ln e k ln k k ln.97 N e.97t e.97t t ln. hours.97 N e kt 8 e k. e k k N eln.t eln.t eln.t ln ln. t t ln. ln. ln. hours. R et8. R 8 et8 8 e t8 8 t 8 ln 8 et8 t 8 ln e t8 t 8 ln 8,8 ears old t 8 ln 797 ears old Ce kt C Ce7k ln 7k k ln 7 The ancient charcoal has onl % as much radioactive carbon..c Celn.7t ln. t ln. 7 t 7 ln. ln., ears.,,788,, 8, m 8,,788 b,788 Linear model: V 9t,788 9 a,788 8,,788e k ek 797 ln k 797 k ln.8 797 (c), The eponential model depreciates faster in the first two ears. CONTINUED Eponential model: V,788e.8t
Chapter Eponential and Logarithmic Functions. CONTINUED (d) t V 9t,788 V,788e.8t $,9 $, $, $,779 (e) The linear model gives a higher value for the car for the first two ears, then the eponential model ields a higher value. If the car is less than two ears old, the seller would most likel want to use the linear model and the buer the eponential model. If it is more than two ears old, the opposite is true..,,, (c) m V t e k ln k k.9 V e.9t (d) t The eponential model depreciates faster in the first two ears. (e) The slope of the linear model means that the computer depreciates $ per ear, then loses all value in the third ear. The eponential model depreciates faster in the first two ears but maintains value longer. V t V e.9t $8 $ $79 $8. St e kt e k S 8 e k 8 e k.8 e k ln.8 ln e k k ln.8 Sales (in thousands of units) 9 Time (in ears) t k. (c) S e.., units St e.t. N e kt N 9, t N 9 e k e.t e k e k ln e k ln k ln k. e.t ln ln e.t ln.t t ln das. So, N e..
Section. Eponential and Logarithmic Models 7..e, 7 8..9. 7 7 The average IQ score of an adult student is. The average number of hours per week a student uses the tutor center is.. 9. pt 9e.t. p animals 9e. 9e.t 9e.t 9e.t S,.e kt,,.e k.e k.e k e k 9 (c) e.t 9 t ln9 months. k ln 9 k ln 9. The horizontal asmptotes are p and p. The asmptote with the larger p-value, p, indicates that the population size will approach as time increases. So, When t 8: S S,.e.t., 87,7 units sold..e.8. R log I log I since I. I. R log I log I since I. I 7.9 log I I 7.9 79,,8 8. log I I 8. 99,, R log 8,, 7.9 R log 8,7, 7.8 (c). log I I.,89 (c) R log,.. log I I where I wattm. log log decibels log log 7 7 decibels (c) log 8 log decibels (d) log log decibels
Chapter Eponential and Logarithmic Functions. I log I where I wattm I log log decibels log log decibels (c) log log 8 8 decibels (d) log log decibels. log I log I I log II I I I I I. % decrease I 9. I 8. I 9. 9% log I I I I I % decrease I 8.8 I 7. I 8.8 97% I 7. ph logh 8. log.. ph logh log..9 9..8 logh.. logh.8 logh. H.8 logh H. mole per liter.8 H H.8 mole per liter..9 logh..9 logh H.9 for the apple juice 8. logh 8. logh H 8 for the drinking water.9 8. times the hdrogen ion concentration of drinking water ph logh ph logh ph H ph H ph H The hdrogen ion concentration is increased b a factor of.. T 7 t ln 98. 7 At 9: A.M. we have: 8.7 7 t ln 98. 7 hours From this ou can conclude that the person died at : A.M.
Section. Eponential and Logarithmic Models. Interest: Principal: u M M Pr r t v M Pr r t P,, t, r.7, M 89.9 (c) P,, t, r.7, M 9.7 8 u 8 u v v. In the earl ears of the mortgage, the majorit of the monthl pament goes toward interest. The principal and interest are nearl equal when t ears. u,.7t.7 t, The interest is still the majorit of the monthl pament in the earl ears. Now the principal and interest are nearl equal when t.79 ears. From the graph, u $, when t ears. It would take approimatel 7. ears to pa $, in interest. Yes, it is possible to pa twice as much in interest charges as the size of the mortgage. It is especiall likel when the interest rates are higher.. t.77.s.87 ln s t.9.s Linear model: t.79s. Eponential model: t.8e.9s t.8.9 s t t t t or (c) s 7 8 9 t t t t...7 9...9 9...9 7. 9...9 9.9..9 7....8 8..7.9. 8.8.8.8. t (d) Model : S... 7.7 9. 9...8.9 9.. Model t : S...9 7 7 9. 9...8.9 9.9. Model t : S...9 7 7. 9....8.8 8.. Model t : S..7.9 7. 9. 8.9.9.8.9.. The quadratic model, t, best fits the data. Note: Table values will var slightl depending on the model used for t.
Chapter Eponential and Logarithmic Functions 7. False. The domain can be the set of real numbers for a logistic growth function. 8. False. A logistic growth function never has an -intercept. 9. False. The graph of f is the graph of g shifted 7. True. Powers of e are alwas positive, so if a >, a upward five units. Gaussian model will alwas be greater than, and if a <, a Gaussian model will alwas be less than. 7. Logarithmic Logistic (c) Eponential (deca) (d) Linear (e) None of the above (appears to be a combination of a linear and a quadratic) (f) Eponential (growth) 7. Answers will var. 7.,,, 7. (, ),,, (, ) (, ) (, ) d (c) Midpoint: (d) m,, 7 d (c) Midpoint: (d) m 9,, 7.,,, 7.,, 7, 8 (, ) 8 8 (, ) (7, ) (, ) 8 d (c) Midpoint: (d) m, 7, d 7 9 (c) Midpoint: (d) m 7 7, 7,
( Section. Eponential and Logarithmic Models 77. (, (, (,,, 78. 7,,, (, ( ( 7, ( d (c) Midpoint: (d) m 8, 8, 8 d 7 (c) Midpoint: (d) m 9. 7,, 7 79. 8. Line Slope: m -intercept:, 8 Line Slope: m -intercept:, 8 8. 8. Parabola Verte:, 7 7 89 8 Parabola Verte: 7, 89 8 -intercepts:,,, 8 8. 8. 8 Parabola Verte: Focus:,, 7 Parabola Verte: Focus: 8,, Directri: 8 Directri:
Chapter Eponential and Logarithmic Functions 8. 8. Vertical asmptote: Horizontal asmptote: Vertical asmptote: Slant asmptote: 8 8 87. 8 88. Circle Center:, 8 Radius: 8 8 8 7 Parabola Verte:, P Focus:,. Directri:.7 7 8 8 89. f 9. f Horizontal asmptote: Horizontal asmptote: f.... 9 f 9 8 8 8 8 9. f Horizontal asmptote: f.99.89.7
Review Eercises for Chapter 7 9. f Horizontal asmptote: f 8 9 9. Answers will var. Review Eercises for Chapter. f.. f. f. f... 7.99 f.78 f..7. f 78. f 78.8 f 7.. f f 7. f.8.8.8.9 7. f 8. f 9. f Intercept:, Intercept:, Intercept:, Horizontal asmptote: -ais Horizontal asmptote: Horizontal asmptote: -ais Increasing on:, Decreasing on:, Decreasing on:, Matches graph (c). Matches graph (d). Matches graph.. f. Intercept:, Horizontal asmptote: Increasing on:, Matches graph. f. g Since g f, the graph of g can be obtained b shifting the graph of f one unit to the right. f, g Because g f, the graph of g can be obtained b shifting the graph of f three units downward.. f. g Since g f, the graph of g can be obtained b reflecting the graph of f about the -ais and shifting f two units to the left. f, g 8 Because g f 8, the graph of g can be obtained b reflecting the graph of f in the -ais and shifting the graph of f eight units upward.. f Horizontal asmptote: 8 f 8...
8 Chapter Eponential and Logarithmic Functions. f 7. f. Horizontal asmptote: Horizontal asmptote: f.. 7 9 f.77. 7. 8. 9 7 8 9 8. f. 9. f Horizontal asmptote: Horizontal asmptote: f...77 7. f.8.. 9 8. f. f Horizontal asmptote: Horizontal asmptote: 7 8 9 f.98. f.. 7 8
Review Eercises for Chapter 9. f 8 Horizontal asmptote: f.87.98. 9. 8. e 7 e. 7 e 8 e 8 7. e 8 98.98 8. e 8.88 9. e.7.8. e.78.. h e. h e h.7...7.7..9. 7. f e. st e t, t > t f.7.7 7.9.9.7..7.. 7 t
Chapter Eponential and Logarithmic Functions. A. n n or A e. n Continuous Compounding A $9.98 $. $9. $9. $7. $7.9. A. n n or A e. n Continuous A $8.88 $8799.8 $888. $89.9 $89. $89.8 7. Ft) e t F. F.87 (c) F.8 8. Vt, t 9. A,e.87 $,9,7., The doubling time is ln 7.9 ears..87 V, $787 (c) According to the model, the car depreciates most rapidl at the beginning. Yes, this is realistic.. Q t. For t : Q. grams For t : Q..79 grams (c) Q Mass of Pu (in grams) 8 8 t Time (in ears)... e.8.... log log ln.....8. e. ln f log. log 9 log 9 9 f log log
Review Eercises for Chapter 7. g log 8. g 8 log 8 log f log 9. f log log 7 log 7 7. log 8 log 8. ln 9 ln. ln ln 9. g log 7 7. g log Domain:, -intercept:, Vertical asmptote: 7 7 9 g Domain:, log -intercept:, Vertical asmptote: g. f log Domain:, -intercept:, Vertical asmptote:.. f. f log Domain:, log log. -intercept:., Vertical asmptote: 8 8 8 f...8.9 7 7. f log Domain:, -intercept: 999, Since log log 999. f.7.... 7 Vertical asmptote:
Chapter Eponential and Logarithmic Functions 8. f log Domain:, log log. -intercept:., 7 8 f....7 7 8 9 Vertical asmptote: 9. ln..8. ln.98.. ln e. ln e 7 7. ln7.. ln 8.. f ln. f ln Domain:, -intercept: ln ln e e, Vertical asmptote: Domain:, ln e -intercept:, Vertical asmptote: 8... f.9....9..9.9 7. h ln ln 8. f ln Domain:,, -intercepts: ±, Vertical asmptote: Domain:, ln ln e -intercept:, ±. ± ± ± ± Vertical asmptote:.9.9..77.7..7..7 9. h loga 7 7. h log 7. inches s 7. miles ln ln 7. log 9 log 9.8 log log 9 ln 9.8 ln
Review Eercises for Chapter 7. log log log log ln ln. 7.. log log. 7. log log ln. ln log.8 log.8 log.8 log ln.8 ln.9.9 7. log 8 log 7. log log. log log log log log log log log.8 77. ln ln 78. ln e ln ln e 79. log log log ln ln.99 ln log.9 8. log 7 log 7 log 8. log 7 log log log log 8. log log log 7 log 7 log 7 log 7 log 7 log log log log 7 log 7 log log 8. ln z ln ln ln z 8. ln ln ln z ln ln ln ln ln ln ln 8. ln ln ln 8. ln ln ln ln ln ln ln ln ln ln ln ln, > 87. log log log 88. log log z log log z log z 89. ln ln ln ln ln 9. ln ln ln ln ln 9. log 8 7 log 8 log 8 log 8 7 9. log 8 7 log log log log log log
Chapter Eponential and Logarithmic Functions 9. ln ln ln ln ln 9. ln ln ln ln ln ln ln ln ln ln ln ln 9. 8, t log 8, h Domain: h < 8, (c) As the plane approaches its absolute ceiling, it climbs at a slower rate, so the time required increases. (d) 8, log. minutes 8,, Vertical asmptote: h 8, 9. Using a calculator gives 97. 8 98. 99. e s 8. ln t. 8 8 ln. e. log. log. ln ln e ln log e ln.79. ln. e. e 7. e e e.98 ln e ln ln e ln ln.8 ln ln.7 or 8. e 9.. 8 8 e ln e ln log log log ln ln. log log.9 or ln ln log ln ln.7
Review Eercises for Chapter. 8. 7 ln ln 7 ln ln 7 ln 7 ln.7 9 9 ln ln 9 ln ln 9 ln 9.8 ln. e 7e. e e e or e ln e ln ln e ln ln.9 ln.9 e e 8 e e e or e ln ln.9.8... Graph.. The -intercepts are at.9 and at 7.8.. Graph.. The -intercepts are at 7.8 and at.7. 9 7. e. 8. Graph e. and. The graphs intersect at.7. 8 e. 9 Graph e. and 9. The graphs intersect at.7. 9. ln 8.. e ln e 8. e 8. e8.. ln 7.. e 7. e7. 7.88 ln ln e ln e 7. e 7. e7... ln. ln e e.7 ln ln. ln e ln e e e.7 ln 8 ln 8 ln 8 8 e e 8 9.9
Chapter Eponential and Logarithmic Functions. ln. ln ln ln ln e ln e e e.98 ln e e 7.99 7. log 8 log 8 log 8 8. log 8 log 8 Since is not in the domain of log 8 or of log 8, it is an etraneous solution. The equation has no solution. log log log log log ±, Quadratic Formula Onl.9 is a valid solution. 9. log..9 log. ln 8. Graph ln and 8. log Graph log. (., 8) 9 9 The graphs intersect at approimatel., 8. The solution of the equation is.. 8 The -intercepts are at,., and.7.. ln Graph ln and. The graphs do not intersect. The equation has no solution.
Review Eercises for Chapter 7. log. Let log. 8 7 7e.7t e.7t ln ln e.7t ln.7t t ln. ears.7 The -intercepts are at.99 and.77.. S 9 logd 7. 8 9 logd 8 9 logd logd 8 9 d 89.8 miles e Eponential deca model Matches graph (e). 8. e 9. Eponential growth model Matches graph. ln Logarithmic model Vertical asmptote: Graph includes, Matches graph (f).. 7 log. Logarithmic model Vertical asmptote: Matches graph (d). e. Gaussian model Matches graph. e Logistics growth model Matches graph (c).. ae b. ae b a e b. e b ln. b b. Thus, e.. ln ae b aeb a eb e b ln b b b. e.
8 Chapter Eponential and Logarithmic Functions. P 99e.t.. million thousand 99e.t e.t 99 ln.t 99 t ln99. 8. ears According to this model, the population of South Carolina will reach. million during the ear 8. Ce kt C Ce,k ln ln e,k ln,k k ln, When t, we have Ce ln,.98c 98.%C. After ears, approimatel 98.% of the radioactive uranium II will remain. 7.,,e r e r 8. N and N so N e kt and: 9..99e 78, ln r e k Graph.99e 78. 7 ln r ek. r.89.89% A,e.89 $,8.98 k ln 7 k ln7.889 The population one ear ago: The average test score is 7. N e.889 bats. 7 N.e.t When When N :.e.t 7.e.t 7 e.t 7 7.t ln 7 7 7.e.t t ln77. 7.7 weeks N 7: 7.e.t 7 7.e.t 8 7 e.t 8.t ln 8 7.e.t t ln8.. weeks
Problem Solving for Chapter 9. log I log I. log I. I I. wattcm. R log I since I. log I 8. I 8.,88, log I.8 I.8 7,79,8 (c) log I 9. I 9.,8,9,. True. B the inverse properties, log b b.. False. ln ln ln ln. Since graphs and (d) represent eponential deca, b and d are negative. Since graph and (c) represent eponential growth, a and c are positive. Problem Solving for Chapter. a.... 7 The curves. and. cross the line. From checking the graphs it appears that will cross a for a.. e The function that increases at the fastest rate for large values of is e. (Note: One of the intersection points of e and is approimatel., 9 and past this point e >. This is not shown on the graph above.). The eponential function, e, increases at a faster rate than the polnomial function n.. It usuall implies rapid growth.. fu v a uv. a u a v fu fv f a a f f g e e e e e e e e 7. = e = e (c) = e
Chapter Eponential and Logarithmic Functions 8.!!!! = e 9. f e e e e e e e e As more terms are added, the polnomial approaches e. e e e e e!!!!!... e ± Quadratic Formula Choosing the positive quantit for e we have ln. Thus, f ln.. f a, a a >, a a a a a a a a a log a ln ln a f. Answer (c). e The graph passes through, and neither nor pass through the origin. Also, the graph has -ais smmetr and a horizontal asmptote at.. The steeper curve represents the investment earning compound interest, because compound interest earns more than simple interest. With simple interest there is no compounding so the growth is linear. Compound interest formula: Simple interest formula: A.7 t.7 t A Prt P.7 t (c) One should choose compound interest since the earnings would be higher. Growth of investment (in dollars) A Compounded Interest Simple Interest t Time (in ears). and c tk c tk c tk c c tk tk ln c c t k t k ln ln c ln c t k k ln t c tk ln c ln c k k ln. B B a kt through, and, log a B a k ak k log a k B a log at a log a t t
Problem Solving for Chapter. (c).. t. Let log a m and log ab n. Then a m and.88t.t 9,78 ab n.,9, a m a b n 8, (d) Both models appear to be good fits for the data, but neither would be reliable to predict the population of the United States in. The eponential model approaches infinit rapidl. a mn a b a mn b log a b m n log a b m n log a b log a log ab 7. ln ln ln ln ln ln ln or ln or e 8. ln (c) = ln = ln = ln 9 9 9 9. The pattern implies that ln.... = ln 9. ab a b. ln lnab ln lna b 8. ln ln ln a ln b ln ln a ln b ln ln b ln a Slope: m ln b -intercept:, ln a ln ln a ln b ln ln a b ln ln b ln ln a Slope: m b -intercept:, ln a 8. ln 7.7 ft min
Chapter Eponential and Logarithmic Functions. cubic feet per minute (c) Total air space required: 8, cubic feet 8. ln Let floor space in square feet and h feet. ln. V h ln., 8 e. 8 cubic feet of air space per child. If the ceiling height is feet, the minimum number of square feet of floor space required is 8 square feet.. 9. The data could best be modeled b a logarithmic model. (c) The shape of the curve looks much more logarithmic than linear or eponential. (d) 9.8.7 ln The data could best be modeled b an eponential model. (c) The data scatter plot looks eponential. (d) 9.. 9 9 (e) The model is a good fit to the actual data. 9 (e) The model graph hits ever point of the scatter plot.. 9. 9 9 The data could best be modeled b a linear model. (c) The shape of the curve looks much more linear than eponential or logarithmic. (d).788 8. 9 The data could best be modeled b a logarithmic model. (c) The data scatter plot looks logarithmic. (d).99.9 ln 9 (e) The model is a good fit to the actual data. 9 (e) The model graph hits ever point of the scatter plot.
Practice Test for Chapter Chapter Practice Test. Solve for : 8.. Solve for : 8.. Graph f.. Graph g e.. If $ is invested at 9% interest, find the amount after three ears if the interest is compounded monthl. quarterl. (c) continuousl.. Write the equation in logarithmic form: 7 9. 7. Solve for : log. 8. Given log evaluate log b. and log b.87, b 8. 9. Write ln ln ln z as a single logarithm.. Using our calculator and the change of base formula, evaluate log 9 8.. Use our calculator to solve for N: log N.. Graph log.. Determine the domain of f log 9.. Graph ln.. True or false: ln ln ln. Solve for : 7. Solve for : log 8. Solve for : log log 9. Solve for : e e. Si thousand dollars is deposited into a fund at an annual interest rate of %. Find the time required for the investment to double if the interest is compounded continuousl.