CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs......... Section. Properties of Logarithms.................. Section. Solving Eponential and Logarithmic Equations..... Section. Eponential and Logarithmic Models........... Section. Nonlinear Models..................... Review Eercises............................. Practice Test...............................
CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs You should know that a function of the form a, where a >, a, is called an eponential function with base a. You should be able to graph eponential functions. You should be familiar with the number e and the natural eponential function f e. 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.............. g. f Asmptote: Intercept:, Increasing Asmptote: Intercept:, Increasing
Chapter Eponential and Logarithmic Functions. f. h Asmptote: Intercept:, Decreasing Asmptote: Intercept:, Decreasing. h. g Asmptote: Intercept: Increasing, Asmptote: Intercept:, Increasing. g. f. f rises to the right. Asmptote: Intercept:, Matches graph (d). Asmptote: Intercepts:,,., Decreasing Asmptote: Intercept:, Decreasing. f is positive and decreasing. Matches graph.
Section. Eponential Functions and Their Graphs. f rises to the right. Asmptote: Intercept:, Matches graph (c).. f is increasing and has, intercept. Matches graph.. f. f g f g f Horizontal shift five units to the right Vertical shift five units upward. f g f Horizontal shift four units to the left, followed b reflection in -ais. f. g. f Reflection in -ais followed b vertical shift five units upward. f. g f Horizontal shift two units to the right followed b vertical shift three units downward f g Reflection in the -ais followed b left shift of four units. e... e e. e...
Chapter Eponential and Logarithmic Functions..e.. f. f f.... f.... Asmptote: Asmptote:. f. f f.. f.. Asmptote: Asmptote:. f f. Asmptote:. f f... Asmptote:
Section. Eponential Functions and Their Graphs.......... Asmptote: Asmptote:...... Asmptote: Asmptote:. f e. s t e.t f.... Asmptote: t s t..... Asmptote: t
Chapter Eponential and Logarithmic Functions. f e. f e. f.... f.... Asmptote: Asmptote:. f e. g e f.... g.... Asmptote: Asmptote:. s t e.t. g e t s t.... Asmptote: t g.... Asmptote:
Section. Eponential Functions and Their Graphs. f e. f.. Horizontal asmptotes:,... f....... f undef.... Horizontal asmptote: Vertical asmptote:. f e... f... f.... Horizontal asmptotes:, Vertical asmptote:... f Asmptotes:....... ln. f undef...... ln Intersection:.,
Chapter Eponential and Logarithmic Functions.,. f e Intersection:.,, Decreasing:,,,. Increasing on and,, f e Decreasing on, (c) Relative maimum:,. Relative minimum:, Increasing:, (c) Relative maimum:, e,. Relative minimum:,. P, r.%., t Compounded n times per ear: A P n r nt. n n Compounded continuousl: A Pe rt e. n Continuous A....... P, r %., t n Continuous A....... P, r %., t Compounded n times per ear: A P r n nt. n n Compounded continuousl: A Pe rt e. n Continuous A....... P, r %., t n Continuous A......
Section. Eponential Functions and Their Graphs. P,, r %. A Pe rt e.t t A,.,.,.,.,.,.. P,, r %., compounded continuousl: A Pe rt,e. t t A,.,.,.,.,.,.. P,, r.%. A Pe rt,e.t t A,.,.,.,.,.,.. P,, r.%., compounded continuousl: A Pe rt,e.t t A,.,.,.,.,.,... A..... $. A... $,. A.. $. A.. $.. p e. If, p $.. (c) For, p $.. p.......
Chapter Eponential and Logarithmic Functions.... e. Q t When t, Q grams. When t, Q. grams. (c) has the highest return. After ears,.. $... $... $. (d) Never. The graph has a horizontal asmptote Q.. Q t. P t e.t When t, Q. When t, Q. grams. (c) Q Mass of C (in grams) Time (in ears) t (c) P P P P e. P e.. P e... (c) P.e.t e.t. (Answers will var.).t ln. t, or Year P........ Year P........
Section. Eponential Functions and Their Graphs. C t P. t., C. (c) C... t V t,,, V (c) According to the model, V t as t increases. However, V.. True. f is not an eponential function.. False. e is an irrational number.. The graph decreases for all and has positive -intercept. Matches (d).. e e increases at the fastest rate. For an positive integer n, e > n for sufficientl large. That is, e grows faster than n. (c) A quantit is growing eponentiall if its growth rate is of the form ce r. This is a faster rate than an polnomial growth rate.. f. and g e... (c) and (d) are eponential functions because the eponents are variable. (Horizontal line).. f g f e > e As, f g. e., e...,.. >, >, >
Chapter Eponential and Logarithmic Functions. f has an inverse because f is one-to-one. f. f is one-to-one, so it has an inverse. f f. f has an inverse because f is one-to-one.. f is not one-to-one, so it does not have an inverse. f. f Vertical asmptote: Horizontal asmptote: Intercept:,. f Slant asmptote: Vertical asmptote: Intercept:,. Answers will var.
Section. Logarithmic Functions and Their Graphs 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. (c) log a a since a a. log a a since a a. (d) 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. (c) ln e since e e. ln e since e e. (d) If ln ln, then. You should be able to graph logarithmic functions. Vocabular Check. logarithmic function.. natural logarithmic. a log a.. log. log. log. log. log. log. ln e. ln..... ln e e e e..... ln e e e. ln e e e. ln e e e. log. log. log. log.. log.. log. e.... ln..... e.... ln..... e..... ln...... e..... ln...... e.... ln....
Chapter Eponential and Logarithmic Functions. e e.... ln..... log log. log because.. g log. log g, log, log. log.. log.. log.... log... log log. log. log. log. log log. log log. log log. log. log log. log log. f and g log are inverses of each other.. f and g log are inverses of each other. f g f. f e and g ln are inverses of. f and g log are inverses of each other. each other. f g f g g
Section. Logarithmic Functions and Their Graphs. log. log Domain: > > Vertical asmptote: log -intercept:, Domain: > > Vertical asmptote: log -intercept:,. log. log Domain: > Vertical asmptote: log log -intercept:, Domain: > Vertical asmptote: log log -intercept:,. log. log. Domain: > > Vertical asmptote: log log -intercept:, Domain: > > Vertical asmptote: log log -intercept:, f log. f log. f log Asmptote: Asmptote: Asmptote: Point on graph:, Point on graph:, Point on graph:, Matches graph. Matches graph (c). Matches graph (d).
Chapter Eponential and Logarithmic Functions. f log Asmptote: Domain: > < Point on graph:, Matches graph.. f log g log is a reflection in the -ais of the graph of f.. The graph of g log is a horizontal shift units to the left of the graph of f log.. f log. The graph of g log is a vertical shift g log is obtained from f b a reflection in the -ais followed b a vertical shift four units upward. three units upward of the graph of f log.. Horizontal shift three units to the left and a vertical shift two units downward. Horizontal shift one unit to the right and a vertical shift four units upward. ln.. ln... ln.. ln... ln e. ln e. e ln... (Inverse Propert) (Inverse Propert) ln e ln. f ln Domain: > Vertical asmptote: -intercept:,. h ln ln e Domain: > > The domain is,. Vertical asmptote: -intercept: ln The -intercept is,. e....
Section. Logarithmic Functions and Their Graphs. g ln Domain: > < The domain is,. Vertical asmptote: -intercept: ln e The -intercept is,.. f ln Domain: > < The domain is,. Vertical asmptote: -intercept: ln e The -intercept is,. ln e..... g ln is a horizontal shift three units. g ln is a horizontal shift four units to the left. to the right.. g ln is a vertical shift five units. g ln is a vertical shift four units downward. upward.. g ln is a horizontal shift one unit. g ln is a horizontal shift to the right and a vertical shift two units upward. two units to the left and a vertical shift five units downward.. f( ln Domain:, (c) Increasing on, Decreasing on, (d) Relative minimum:,.. g ln Domain:, (c) Increasing on,. Decreasing on., (d) Relative maimum:.,.
Chapter Eponential and Logarithmic Functions. h ln. f ln Domain:, (c) Increasing on., Decreasing on,. (d) Relative minimum:.,. Domain:,, (c) Increasing on., Decreasing on,,. (d) Relative minimum:.,.. f ln. f ln Critical numbers:, > ; Test intervals:,,,,, Testing these three intervals, we see that the domain is,,. (c) The graph is decreasing on, and decreasing on,. (d) There are no relative maimum or minimum values. Critical numbers:, > ; Test intervals:,,,,, Testing these three intervals, we see that the domain is,,. (c) The graph is increasing on, and increasing on,. (d) There are no relative maimum or minimum values.. f ln. > ; Domain: all (c) The graph is increasing on, and decreasing on,. (d) There are no relative maimum or relative minimum values. f ln Domain: > (c) The graph is increasing on, and decreasing on,. (d) Relative maimum:,.
Section. Logarithmic Functions and Their Graphs. f ln. f ln ln ; Domain: (c) The graph is increasing on,. (d) There are no relative maimum or relative minimum values. Domain: > (c) The graph is decreasing on, and increasing on,. (d) Relative minimum:,. f t log t, t f log f log. (c) f log. (d). The model is a good fit. T > F when p >. pounds per square inch The graph of T and intersect at p.. (c) T. F.. t ln K. K t...... As the amount increases, the time increases, but at a lesser rate. log I. I : log log decibels I : log log decibels (c) No, this is a logarithmic scale. r... t. r. r. r The doubling time decreases as r increases. r... t. r. r. r.
Chapter Eponential and Logarithmic Functions. t. ln, >. ln. ears... ln. ln. ft min. ln. ears..,..,. Interest for -ear loan is,.,,.. Interest for -ear loan is,.,,.... ln, cubic ft per minute cubic feet per minute per child children From the graph, for ou get cubic feet. (c) If ceiling height is, then square feet of floor space is needed.. False. You would reflect in the line.. True. log log. log b. b b log b. b b log b. b b log b b b. The vertical asmptote is to the right of the -ais, and the graph increases. Matches.. The vertical asmptote is to the left of the -ais. Matches.. f log is the inverse of g a a, where a >, a.. f ln, g. False, is not an eponential function of. ( can never be.) f ln, g f g g f True, could be log. (c) True, could be. (d) False, is not linear. (The points are not collinear.), The rate of growth of the natural logarithmic function is slower than for an n. g n
Section. Logarithmic Functions and Their Graphs. = ln Pattern is.... = ln As ou use more terms, the graph better approimates the graph of ln on the interval,.. f ln f..... As increases without bound, f approaches. (c).. f t ln t ln t ln t t e. Or, ou could graph f t and together in the same viewing window, and determine their point of intersection... ± ± i i i....
Chapter Eponential and Logarithmic Functions... f g f g. f g. fg f g. g f.. and intersect at. The graphs of and intersect when. or.. The graphs of and intersect when. or.. and do not intersect. No solution Section. Properties of Logarithms You should know the following properties of logarithms. log a log b log b a log a uv log a u log a v ln uv ln u ln v (c) log a u v log a u log a v ln u v ln u ln v (d) log a u n n log a u ln u n n ln u You should be able to rewrite logarithmic epressions using these properties. Vocabular Check. change-of-base. ln ln a. log a u n. ln u ln v. log log log log ln ln. log log log log ln ln
Section. Properties of Logarithms. log log log log log log ln ln ln ln. log log log log log log ln ln ln ln. log a log log a log a ln ln a. log a log log a log a ln ln a. log. log log. log. ln ln.. log. log log. log. ln ln.. log ln. ln. log ln. ln. log ln. ln log ln ln. log. ln. ln. ln ln. log. ln. ln.. log ln ln. ln. log. ln. ln ln. ln ln ln ln. ln ln ln ln ln ln ln ln ln ln ln. ln ln ln. log b log b.. ln ln ln ln log b log b. log b log b log b.... log b log b log b log b log b... log b.. log b log b..
Chapter Eponential and Logarithmic Functions. f log ln ln. f log ln ln. f log ln ln ln ln. f log ln ln ln ln. f log ln ln ln ln. f log ln ln ln ln. log log log. log log log log log log log log log log log log. ln e ln ln e ln ln. log log log log log log log log log log. log log log.. ln ln ln e e ln ln e ln ln ln ln ln ln ln ln ln. log z log log z log z. log log log. log. log log log log
Section. Properties of Logarithms. log. ln z ln z z log ln z z. ln t ln t ln t. ln z ln ln ln z. ln z ln ln ln z. log a bc log a log b log c. log a log b log c log z log log log z log log log z. ln a a ln a ln a. ln z z ln z ln z ln a ln a, a > ln z ln z. ln ln. ln ln ln ln ln ln ln ln ln ln ln 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 ln z ln ln ln z ln ln (c) The graphs and table suggest that for >. In fact, ln ln ln ln ln. log b z log b log b z log b log b log b z log b log b log b z..............
Chapter Eponential and Logarithmic Functions. ln, ln ln.......... (c) The graphs and table suggest that. In fact, ln ln ln ln ln.. ln ln ln.. ln ln z ln z log z log log z. log log t log t. log log. log z log z. ln ln. ln ln ln ln ln ln ln. ln ln ln ln. ln ln ln ln ln ln. ln ln ln.. ln ln ln ln ln ln ln ln ln ln ln z ln z ln z ln z ln z. ln z z ln z ln z z z ln ln ln z ln ln ln z ln ln z ln z
Section. Properties of Logarithms. ln ln ln ln ln ln ln ln ln ln. ln ln ln ln ln ln ln ln. ln ln ln ln ln ln ln ln ln ln ln.. ln ln ln ln ln ln ln ln ln.............. ln ln ln ln ln ln ln (c) The graphs and table suggest that. In fact, ln ln ln ln.
Chapter Eponential and Logarithmic Functions. ln ln, ln, > ERROR.... ERROR.... (c) The graphs and table suggest that. In fact, ln ln ln ln ln.. ln ln The domain of is >..... undefined undefined.. (c) The graphs and table suggest that for >. The functions are not equivalent because the domains are different.. ln, ln ln.. ERROR... ERROR ERROR ERROR... (c) No, the epressions are not equivalent. The domain of is all, whereas the domain of is >.. log log. log.. log. log.. log is undefined. is not in the domain of f log.. log log log log log. ln e ln e. log log log. log log log. log is undefined because is not in the domain of log.. log log log log log log. ln e ln e ln e ln e. ln e ln e. ln e.. ln e.
Section. Properties of Logarithms. ln e ln ln e ln e. ln e ln e ln e. (c) log I log I log log I log log I log I I log log log log log log. f t log t, t When t, f. (c) f (d) f (e) f t when t months.. T.. t T.. t CONTINUED The data t, T fits the model T.. t. The model T.. t fits the original data.
Chapter Eponential and Logarithmic Functions. CONTINUED (c) ln T.t., linear model (d). T e.t. T.e.t.. t.t., T linear model T.t. T.t.. If ab, then ln ln ab ln a ln b, which is linear. If then c d. c d,. True. False. For eample, let and a.. False. For eample, let and a. Then f a ln, but Then But f ln. f f a ln ln ln. f a ln. f a ln. False. For eample, let and a. Then f a ln ln, but f f a ln ln.. False. ln ln In fact, ln ln.. False. For eample, let n and e.. True. In fact, if ln <, then < <. Then f n ln e, but nf ln e.. False. For eample, let e. Then f ln e ln e >, but e < e.. Let and, then a a log a z log a b and b z a z log a b a z z b z z b z z log a b log a log a b.
Section. Properties of Logarithms. f ln g f = h. f log ln ln g ln ln h ln ln f h b Propert.. f log ln ln. f log ln ln. f log ln ln. f log ln ln. f log ln ln ln. ln, ln., ln., ln. ln. ln. ln ln ln... ln. ln ln ln... ln ln ln.. ln ln ln.. ln ln ln... ln ln ln ln ln... ln ln ln... ln ln ln.. ln ln ln ln ln... ln ln ln ln ln...
Chapter Eponential and Logarithmic Functions... if,.. ± ±,,..,, ±, ±... ±, ± ±, ±,,
Section. Solving Eponential and Logarithmic Equations Section. Solving 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 a a. ln e 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 Use our graphing utilit to approimate solutions. Vocabular Check. solve. (c) (d). etraneous.. Yes, is a solution. No, is not a solution. No, is not a solution. No, is not a solution.. e e e e + e e No, e is not a solution. ln e ln e ln Yes, ln is a solution. (c). e. e.. e ln e ln e ln Yes. e. e. Yes (c) ln e ln. Yes,. is a solution. No
Chapter Eponential and Logarithmic Functions.. log. is an approimate solution.. No, is not a solution. (c) Yes, is a solution. log ln ln (c).; ln. ln. ; ; log Yes log ;.; ln.. ; No ln No. ln. e. (c) ln. ln e. ln e.. ln ln. ln ln.. Yes, e. is a solution. No, ln. is not a solution.. ln. ln.. Yes,. is a solution.. ln. e. ; ln e. ln e..; Yes ; ln.; Yes (c) ; ln..; No. g f Point of intersection:, Algebraicall:,. f g Point of intersection: g f,
Section. Solving Eponential and Logarithmic Equations. Point of intersection:, Algebraicall: f g,. f. g g f Point of intersection:,, g f Point of intersection:, Algebraicall: log log,. f log ln ln g. g f g f Point of intersection:, Algebraicall: ln e. Point of intersection:, log log, f ln e g Point of intersection:, f g,,
Chapter Eponential and Logarithmic Functions........... log log log. Alternate solution:. log ln ln.. ln ln. ln ln. ln. ln ln ln ln ln e e e... log. log. log.. ln e e.. log ln e e.. ln e ln e. ln e. e ln. e ln. ln e. e ln
Section. Solving Eponential and Logarithmic Equations... t. ln ln ln ln ln ln ln ln t ln ln ln ln ln ln t ln ln ln ln ln ln. t. t. t... ln t ln. t ln ln. t ln. ln ln. t. ln ln ln ln ln ln ln ln ln ln ln. ln ln.... t.. t t ln. ln t t. ln ln...,.... ln. ln. ln. ln... t t ln. ln..,.. ln. ln ln t ln.... ln ln..
Chapter Eponential and Logarithmic Functions. e. e. e e ln ln ln ln.. e e ln ln ln.. e. e. e e e e ln e ln e e ln ln ln e ln ln. ln.. e e. e e. e., e e e e e. e or e e or e. ln ln. e is impossible. ln. or ln. ln... e.,. e e. e e e.. ln ln..,,. e e. ± ±.,. e e,. e e e ln ln ln.
Section. Solving Eponential and Logarithmic Equations. e. e e ln ln ln. e. e. e. e.. ln ln.... e. e.. e.. e... ln. ln... e..... f....... e. e...... e...... f. e e.
Chapter Eponential and Logarithmic Functions.. t t... t. t The zero of. t is t..... e e The zero of e is.. The zero of e is.. g e Zero at.. f e. g t e.t. The zero is.. h t e.t The zero is t.. Zero at t.. ln. ln. ln. e. e. e e.. e.. ln.. ln. ln e. ln ln. e.. log log. ln e e. log log. ln. e.. log z z z
Section. Solving Eponential and Logarithmic Equations. log. log.. log ± ± log.. log... ln. ln. ln e ln e ln e e ln e e. e e or e e e.,. or e.. ln. log log e ± e ±. log log. log log. log is etraneous. ln ln ln ln ln or Both of these solutions are etraneous, so the equation has no solution.
Chapter Eponential and Logarithmic Functions. ln ln ln, > ln ln ± Taking the positive solution, ±... log log log ± ± Choosing the positive value, we have. and... log log, > log, Quadratic in ± Taking the positive root and squaring,.. ±. ln. f......
Section. Solving Eponential and Logarithmic Equations.. log. f..... f........ f....... log Graphing log, ou obtain two zeros,. and... Solving log, ±.. ln ln. Graphing ln ln, ou obtain one zero,.. ln ln Graphing ln ln, ou obtain one zero,... ln ln Graphing ln ln, ou obtain... ln ln Solving ln ln,..... Intersection:., e. Intersection:.,. ln.. The graphs intersect at,.,. e From the graph, we have,.,. Intersection:.,.
Chapter Eponential and Logarithmic Functions... ln ln e e e since e, The graphs intersect at.,... e e. e e e e since e since e,. e e. ln e ln since e ln since > ln e.. ln. ln ln > ln e. ln ln e. e. ln. ln ln since > ln e e. e.t e.t ln.t t ln. ears. e.t e.t ln.t t ln. ears.
Section. Solving Eponential and Logarithmic Equations. e.t. e.t ln.t t ln. ears. e.t e.t ln.t t ln. ears. e.t e.t ln.t t ln. ears. e.t e.t ln.t t ln. ears.. e.t. p. e. e.t p ln.t. e. t ln e. ears.. ln e.t units e.t p ln.t. e. t ln. ears. e.. ln units. p e When p $: When p $:... e..e...e. e. ln ln e. ln. ln. e. e. units.. e...e...e. e. ln ln e. ln. e. e. units
Chapter Eponential and Logarithmic Functions.. ln t. ln t ln t. t., or. V.e. t, t > (c)..e. t. e. t. As t, V.. Horizontal asmptote:. The ield will approach. million cubic feet per acre. ln. t t.. ears ln.. P. e.n f m From the graph we see horizontal asmptotes at and. These represent the lower and upper percent bounds. (c) Males: e.... ln.. Females: e.. e.. e.... ln.. e... inches e... inches Horizontal asmptotes:,. The upper asmptote,., indicates that the proportion of correct responses will approach. as the number of trials increases. (c) When P % or P.:.. e.n e.n.. e.n.. ln e.n ln...n ln... ln. n trials.
Section. Solving Eponential and Logarithmic Equations. T h We see a horizontal asmptote at. This represents the room temperature. (c) h h h h ln ln h ln h ln ln ln h h. hour.,. ln t,. ln t ln t. t., or (c) Let,. ln t and. The graphs of and intersect at t... False. The equation e has no solutions.. False. A logarithmic equation can have an number of etraneous solutions. For eample ln ln ln has two etraneous solutions, and.. Answers will var.. f log a, g a, a >. a. The curves intersect twice: If f log a a g intersect eactl once, then log a a a. f g The graphs of and a intersect once for a e e.. Then log a e e e e e..,. and.,. For a e e, the curves intersect once at e, e. (c) For < a < e e the curves intersect twice. For a > e e, the curves do not intersect.
Chapter Eponential and Logarithmic Functions. Yes. The doubling time is given b P Pe rt e rt ln rt t ln r. The time to quadruple is given b P Pe rt e rt ln rt t ln r ln r which is twice as long. ln r ln r. To find the length of time it takes for an investment P to double to P, solve P Pe rt ln rt ln r e rt t. Thus, ou can see that the time is not dependent on the size of the investment, but rather the interest rate.. f. f. f. f. f, <., f,, >
Section. Eponential and Logarithmic Models Section. Eponential and Logarithmic Models You should be able to solve compound interest problems.. A P r n nt. A Pe rt 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 logistics growth model a be. c d You should be able to use the logarithmic models ln a b and log a b. Vocabular Check. iv i (c) vi (d) iii (e) vii (f) ii (g) v. Normall. Sigmoidal. Bell-shaped, mean. e. e. This is an eponential growth model. Matches graph (c). This is an eponential deca model. Matches graph (e). log This is a logarithmic model, and contains,. Matches graph.. e. ln. e Gaussian model This is a logarithmic model. Logistics model Matches. Matches graph (d). Matches (f ).. Since A,e.t, the time to double is. Since A e.t, the time to double is given b given b,,e.t e.t ln.t t ln. ears.. Amount after ears: A,e. $,. e.t e.t ln.t t ln. ears.. Amount after ears: A e. $.
Chapter Eponential and Logarithmic Functions. Since A e rt and A, when t,. Since A e rt and A when t, we we have the following. have the following., e r e r ln r r ln Amount after ears:..% A e. $,. e r e r ln r r ln..% Amount after ears: A e. $.. Since A e rt and A. when t, we have the following.. e r. ln. r The time to double is given b, e.t e.t er ln.t r ln...% t ln. ears... Since A e rt and A. when t, we have the following.. e r. The time to double is given b e.t ln.t er ln. r e.t r ln...% t ln. ears... Since A Pe.t and A, when t, we have the following., Pe., e. P,. The time to double is given b,.,.e.t e.t ln.t t ln. ears... Since A Pe.t and A when t, we have the following. Pe. P $. e. The time to double is given b..e.t e.t ln.t t ln. ears..
Section. Eponential and Logarithmic Models. P Pe rt ln rt ln r e rt t r % % % % % % t ln r........ ln P P r t r % % % % % % r t ln ln r t ln t ln r t ln ln r...... ln r t... Continuous compounding results in faster growth. A. t and A e.t A. t A. t From the graph, % compounded dail grows faster than % simple interest.. C Cek. ek k ln Ce kt e ln. g C Cek. ek k ln Ce kt. Ce ln. C. C. g C Cek ek k ln Ce kt e ln. g
Chapter Eponential and Logarithmic Functions. C Cek, e,k. ae b. ae b a e b ae b aeb a k ln,. Ce ln, C. C. g ln b ln b b. Thus, e.. eb e b ln b b ln. Thus, e..., a., e b b ln ln. e. ln b ln ae b ae b a eb b b. Thus, e... Australia:,.,,. a. and..e b b..e.t For,. million. Canada:,.,,. a. and..e b b..e.t For,. million. Philippines:,.,,. a. and..e b b..e.t For,. million. South Africa:,.,,. a. and..e b b..e.t For,. million. CONTINUED
Section. Eponential and Logarithmic Models. CONTINUED Turke:,.,,. a. and..e b b..e.t For,. million. The constant b gives the growth rates. (c) The constant b is negative for South Africa.. P.e.t Decreasing because the eponent is negative. For, t and P, people. For, t and P, people. For, t and P, people. (c).e.t t., or..e k k ln. k. For, t and P.e., people.. P.e kt..e k k ln k. For, t and P e., people. Ce kt C Ce k ln k k ln When t, we have Ce ln.c, or.%... Ce kt C Cek ln k k ln V t t, V,,,e k V,e.t m The ancient charcoal has onl % as much radioactive carbon..c Ce ln t ln. ln t V ae kt, V, b, t, V mt b, V, b, V,, m, k ln,., ears (c), (d) The eponential model depreciates faster in the first ear. (e) Answers will var.
Chapter Eponential and Logarithmic Functions. Let t correspond to. V mt b, V b V m m V t t V ae kt, V a V e k k V t e.t ln. (c) (d) The eponential model depreciates faster in the first ear. (e) Answers will var.. S t e kt. (c) e k e k S t e.t k ln. k. S e.., units S e k in hundreds S. in thousands. e k. e k e k. k ln. k. S e. When, S e.. which corresponds to units...e,..e... Maimum point is, the average IQ score. About. hours
Section. Eponential and Logarithmic Models. p t e.t p e. animals (c) e. t e.t e.t e.t The horizontal asmptotes are p and p. The population will approach as time increases. t ln. months. e.t, t For t,. (c) As For t,. t, (d) Answers will var., limiting value.. R log I I log I I R. R log I I log I I.,, R log,,. I.,, R log,,. (c) I.,,, (c) R log,.. I log I I, where I watt per square meter.. (c) log log decibels log log decibels log log decibels log I I log log decibels log decibels (c) decibels I
Chapter Eponential and Logarithmic Functions. log I I. log I I I I I I I I I I % decrease I. I. I..% % decrease I. I. I. %. ph log H log.... log H. H H. moles per liter. ph log H ph log H ph H Hdrogen ion concentration of grape Hdrogen ion concentration of milk of magnesia... ph log H ph log H ph H ph H ph H The hdrogen ion concentration is increased b a factor of.. P,, r., M. u M M Pr r t... t v.. t In the earl ears, the majorit of the monthl pament goes toward interest. The interest and principle are equal when t. ears. (c) P,, r., M. u... t v.. t u v..t u,. t, u v u v when t. ears. From the graph, when u,, t. ears. Yes, a mortgage of approimatel. ears will result in about $, of interest.
Section. Eponential and Logarithmic Models. t ln T. At : A.M. we have t ln.. hours. Thus, we can conclude that the person died hours before A.M., or : A.M.. t. ln T (t is A.M.). ln T. ln T T e. T e.. < Hence, the steaks do not thaw out in time.. False. The domain could be all real numbers.. False. See Eample, page.. True. For the Gaussian model, >.. True. See page.. Slope: Matches. Intercepts:,,,. Line with intercepts, and,. Matches... Slope:. Matches (d). Intercepts:,,,. Line with intercepts, and,. Matches (c).. f. f The graph falls to the left and rises to the right. Falls to the left and falls to right.. g. The graph rises to the left and falls to the right.. Answers will var... g.. Rises to left and rises to right
Chapter Eponential and Logarithmic Functions Section. Nonlinear Models You should be able to use a graphing utilit to find nonlinear models, including: Quadratic models Eponential models (c) Power models (d) Logarithmic models (e) Logistic models You should be able to use a scatter plot to determine which model is best. You should be able to determine the sum of squared differences for a model. Vocabular Check. a b. quadratic. ab. sum, squared differences. ab, ae c. Logarithmic model. Linear model. Quadratic model. Eponential model. Eponential model. Logistic model. Quadratic model. Linear model... Logarithmic model Linear model Eponential model... Eponential model Linear model......... Coefficient of determination: Coefficient of determination: Coefficient of determination:... Logarithmic model
Section. Nonlinear Models...... ln... ln Coefficient of determination: Coefficient of determination: Coefficient of determination:...... ln... ln... Coefficient of determination: Coefficient of determination: Coefficient of determination:............ Coefficient of determination: Coefficient of determination: Coefficient of determination:.... Quadratic model: R.t.t. Eponential model: R.. t Power model: R.t. (c) The eponential model fits best. Answers will var. (d) For, t and R. million. For, t and R. million. Answers will var.
Chapter Eponential and Logarithmic Functions. Quadratic model: R.t.t. Coefficient of determination:. Eponential model: R.. t Coefficient of determination:. Power model: R.t. Coefficient of determination:.... (c) The quadratic model fits best. (d) Using the quadratic model: Year Price...... Answers will var.. Linear model: P.t. Coefficient of determination:. (c) Eponential model: P.. t Coefficient of determination:. Power model: P.t. Coefficient of determination:. CONTINUED (d) Quadratic model: P.t.t. Coefficient of determination:. (e) The quadratic model is best because its coefficient of determination is closest to.
Section. Nonlinear Models. CONTINUED (f ) Linear model: Year Population (in millions)...... Power model: Year Population (in millions)...... Eponential model: Year Population (in millions)...... Quadratic model: Year Population (in millions)...... (g) and (h) Answers will var.. h is not in the domain of the logarithmic function. h.. ln p (c) (d) For p., h. km. (e) For h, p. atmospheres... T.t. No, the data does not appear linear. (c) Subtracting from the T-values, the eponential model is.. t. Adding back, T.. t. T.t.t. Yes, the data appears quadratic. But, for t, the graph is increasing, which is incorrect. (d) Answers will var.
Chapter Eponential and Logarithmic Functions.. P..e. (c) This model is a good fit.. S. e t.. Using a graphing utilit, t. or. The model is a good fit.. Linear model:. Quadratic model:.. Cubic model:... Power model:.. Eponential model:.. (c) Year Linear Quadratic Cubic Power Eponential Answers will var. (d) For, and million metric tons.. Linear model:.t. Logarithmic model:. ln t. Quadratic model:.t.t. Eponential model:.. t Power model:.t. CONTINUED
Section. Nonlinear Models. CONTINUED Linear model: Logarithmic model: Quadratic model: Eponential model: Power model: (c) Linear:. Logarithmic:. Quadratic:. (Best) Eponential:. Power:. (d) Linear:. Logarithmic:. Quadratic:. (Best) Eponential:. Power:. (e) Quadratic model is best.. Answers will var.. True. False. Write b as b e ln b. Then, ab ae ln b ae c... Slope: -intercept:, Slope: -intercept:,
Chapter Eponential and Logarithmic Functions..... Slope: -intercept:,... Slope:. -intercept:,,. Review Eercises for Chapter............. e.. e.. e. e... e.. f. Intercept:, Horizontal asmptote: -ais Increasing on:, Matches graph (c).. f. Intercept:, Horizontal asmptote: -ais Decreasing on:, Matches graph. f Intercept:, Horizontal asmptote: -ais Decreasing on:, Matches graph (d). f Intercept:, Horizontal asmptote: Increasing on:, Matches graph.
Review Eercises for Chapter. f. f.. g Intercept:, Horizontal asmptote: Intercept:, Horizontal asmptote: -ais Intercept:,. Horizontal asmptote: Increasing on:, Decreasing on, Decreasing on:,. g...... h e f e h e. Horizontal asmptote: f e. Horizontal asmptote: Horizontal asmptote:. f e. Horizontal asmptote: Horizontal asmptote: f e Horizontal asmptote:
Chapter Eponential and Logarithmic Functions. f e.. f Horizontal asmptotes:, Horizontal asmptotes:,. A Pe rt,e.t t A,.,.,.,.,.,.. r %., A,e.t t A,,,,,,. V t, t., Q t When t, Q grams. When t, Q. grams. (c) For t, V $,. (c) The car depreciates most rapidl at the beginning, which is realistic... ln e. ln e. g lo... log. e e. log ln.... e e e..... log.. log ln.... log. log. log e..... log log log log log log
Review Eercises for Chapter. log. log log. log. log log. g log ln ln Domain: > Vertical asmptote: -intercept:,. g log Vertical asmptote: Intercept:, Domain: >.. f log ln ln. f log Domain: > Vertical asmptote: -intercept:., Vertical asmptote: Intercept:, Domain: >.. ln... ln... ln.. ln... log log. log. f ln. Domain:, Vertical asmptote: -intercept:., log log. f ln Domain:, Vertical asmptote: Intercept:, log
Chapter Eponential and Logarithmic Functions. h ln. f ln Domain: Domain: > Vertical asmptote: -intercept:,, Vertical asmptote: Intercept:,., t log, h h <, (c) The plane climbs at a faster rate as it approaches its absolute ceiling. (d) If h, t log,. minutes.,, Vertical asmptote: h,. t. ln, > For., t ears. For., t, the total amount paid is. $,.. The interest is,.,. $,... log log. log. log log. log log ln. ln log ln. ln. log log log log ln ln... log. log. log log. ln. ln... f log ln ln. f log ln ln
Review Eercises for Chapter. ln ln f log ln ln. f log ln ln ln ln. log b log b. log b log b log b. log b log b log b log b log b log b........ log b log b. ln e ln ln e. ln e ln e log b log b ln ln e ln e.. ln.. log log. log log log log. log log log log. log log log log. log log log log.. log log log. log log log log log log ln ln ln. ln ln ln. log log log.. ln ln ln ln ln ln ln ln ln ln ln ln ln z ln ln ln z ln ln ln z log log z log log z log z
Chapter Eponential and Logarithmic Functions. ln ln ln ln ln ln ln. ln ln ln ln ln. ln ln ln ln ln ln ln. s ln h ln h s.... (c) As the depth increases, the number of miles of roads cleared decreases.. f t log t log t log t t months...... log. log... log log. ln e.
Review Eercises for Chapter. ln. ln. ln e. e e e e e e. e. e. e ln ln. e ln e ln ln ln. ln ln ln ln.... ln ln ln ln ln ln ln ln. ln ln. ln. ln... e. e e ln ln e e e e e. or e ln. or ln. or... e e ln. ln e. ln e. ln ln ln ln.... ln ln. ln ln. ln e e. ln e e. e e e e e ln. e ln. ln. e. e.. ln ln ln e e.
Chapter Eponential and Logarithmic Functions.. ln. ln ln e e. log log log. log log ±. (Other zero is etraneous.) ± log log log log log No solution log (etraneous).. log. e e e since e.. e e. e since e ln ln since > ln e. ln ln ln since > ln e. e. e.t e.t ln.t t ln. ears.
Review Eercises for Chapter. p.e. p p.e..e. e.. ln units.e..e. e.. ln units. e. e. ln Decreasing eponential Matches graph (e). Intercept:, Increasing Matches. Logarithmic function shifted to left Matches graph (f).. log. e. Vertical asmptote: Decreasing Gaussian model Matches graph. Matches (d). e Logistic model Matches (c).. ae b ae b a e b. e b ln. b b. Thus, e..... ae b ae b a e b eb b ln b ln ln. e. ae b. ae b a eb e b ln b b. Thus, e.. ln ae b ae b a a eb eb b b ln e..
Chapter Eponential and Logarithmic Functions. P e kt t corresponds to., : e k e k k ln k ln ln. P e.t For, P e.. or, population in.. P Pek, ln ln,k k,. After ears, A e k. or.% remains..,,e r.. e r ln r r ln. or.%,e. $,..e The average score corresponds to the maimum,...e.t.e.t e.t..t ln.e.t Similarl: t ln.. R log I I log I I R I.,, I.,,. weeks.e.t.e.t e.t..t ln. t. weeks (c) I.,,,
Review Eercises for Chapter. Logistic model. Linear model. Logarithmic model. Quadratic model (or eponential). Linear model:.t ;. Quadratic model:.t.t ;. Eponential model:.. t ;. Logarithmic model:. ln t ;. Power model:.t. ;. Linear model: Quadratic model: Eponential model: Logarithmic model: Power model: (c) The eponential model is best because its coefficient of determination is closest to. Answers will var. (d) For, t and $ million. (e) when t., or.. Linear model:.t ;. Quadratic model:.t.t ;. Eponential model:. t ;. Logarithmic model:. ln t;. Power model: t. ;. CONTINUED
Chapter Eponential and Logarithmic Functions. CONTINUED Linear model: Quadratic model: Eponential model: Logarithmic model: Power model: (c) The quadratic model is best because its coefficient of determination is closest to. (d) For, t and thousand. (e) when t., or.. P..e., (c) The model is a good fit. (d) The limiting size is., fish.. P.e.t Year P..... Year P...... Year P.... Linear model (c) Slope is.. The population increases b, people each ear. (d).. t. Slope...t., Answers will var..
Review Eercises for Chapter. True; b the Inverse Properties, log b b.. False; ln ln ln ln.. e e e e e True (b Properties of eponents). ln ln False ln ln ln ln. False. The domain of f ln is >.. True. ln ln ln. Since < <, < < < <.. Pattern n i i i! = e!!! The graph of closel approimates e near,.
Chapter Eponential and Logarithmic Functions Chapter Practice Test. Solve for :. Solve for :. Graph f b hand.. Graph g e b hand.. If $ is invested at % interest, find the amount after three ears if the interest is compounded monthl. quarterl. (c) continuousl.. Write the equation in logarithmic form:. Solve for : log. Given log evaluate log b. and log b., b.. Write ln ln ln z as a single logarithm.. Using our calculator and the change of base formula, evaluate log.. Use our calculator to solve for N: log N.. Graph log b hand.. Determine the domain of f log.. Graph ln b hand.. True or false: ln ln ln. Solve for :. Solve for : log. Solve for : log log. Solve for : e e. Si thousand dollars is deposited into a fund at an annual percentage rate of %. Find the time required for the investment to double if the interest is compounded continuousl.. Use a graphing utilit to find the points of intersection of the graphs of ln and e.. Use a graphing utilit to find the power model a b for the data,,,,,, and,.