CHAPTER 3 Exponential and Logarithmic Functions
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1 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
2 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: Asmptote: Intercept:, Intercept:, Increasing Decreasing
3 Chapter Eponential and Logarithmic Functions. h. g Asmptote: Intercept: Increasing, Asmptote: Intercepts:,,., Decreasing. f rises to the right. Asmptote: Intercept:, Matches graph (d).. f rises to the right. Asmptote: Intercept:, Matches graph (c).. f. g f Horizontal shift five units to the right f g f Horizontal shift four units to the left, followed b reflection in -ais. f g f Horizontal shift two units to the right followed b vertical shift three units downward. e... e... f. f f.... f.. Asmptote: Asmptote:
4 Section. Eponential Functions and Their Graphs. f. f..... Asmptote: Asmptote:.. f e... f.... Asmptote: Asmptote:. f e. f e f.... f.... Asmptote: Asmptote:
5 Chapter Eponential and Logarithmic Functions. s t e.t t s t.... Asmptote: t. f e. f.. Horizontal asmptotes:,. f e... f... f.... Horizontal asmptotes:, Vertical asmptote:... f e Intersection:., Decreasing:,,, Increasing:, (c) Relative maimum:, e,. Relative minimum:,
6 Section. Eponential Functions and Their Graphs. 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 Compounded n times per ear: A P r n nt. n n Compounded continuousl: A Pe rt e. n Continuous A P,, r %. A Pe rt e.t t A,.,.,.,.,.,.. P,, r.%. A Pe rt,e.t t A,.,.,.,.,.,.. A..... $. A.. $,.. p. e (c) For, p $.. p If, p $..
7 Chapter Eponential and Logarithmic Functions. Q t. When t, P t e.t When t, (c) Q grams. Q. grams. (d) Never. The graph has a horizontal asmptote Q. (c) P P P P e.. P e. P e... C t P. t C. (c) C.... True. f is not an eponential function.. The graph decreases for all and has positive -intercept. Matches (d).. f. and g e.. (Horizontal line) f g f As, f g.. e., e.. e > e.,. >. f has an inverse because f is one-to-one.. f has an inverse because f is one-to-one. f f
8 Section. Logarithmic Functions and Their Graphs. f Vertical asmptote: Horizontal asmptote: Intercept:,. Answers will var. 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 (c) log a a since a a a since 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. ln e. ln e e e. ln e e e. log. log. log. e.... ln..... e..... ln.....
9 Chapter Eponential and Logarithmic Functions. e.... ln..... log log. g log log. log.. log... log log. log. log log log. log log. log log. f and g log are inverses of. f e and g ln are inverses of each other. each other. f g f g. log. log. log Domain: > > Vertical asmptote: log -intercept:, Domain: > Vertical asmptote: log -intercept: log, Domain: > > Vertical asmptote: log log -intercept:,. f log. f log. f log Asmptote: Point on graph:, Matches graph. Asmptote: Point on graph:, Matches graph (d). g log is a reflection in the -ais of the graph of f.
10 Section. Logarithmic Functions and Their Graphs. f log g log is obtained from f b a reflection in the -ais followed b a vertical shift four units upward.. Horizontal shift three units to the left and a vertical shift two units downward. ln.. ln.. ln e. (Inverse Propert) e ln.. (Inverse Propert). f ln Domain: > Vertical asmptote: -intercept:,. g ln Domain: > < The domain is,. Vertical asmptote: -intercept: ln e The -intercept is,.. g ln is a horizontal shift three units. g ln is a vertical shift five units to the left. downward.. g ln is a horizontal shift one unit to the right and a vertical shift two units upward.. f( ln. h ln Domain:, (c) Increasing on, Decreasing on, (d) Relative minimum:,. Domain:, (c) Increasing on., Decreasing on,. (d) Relative minimum:.,.
11 Chapter Eponential and Logarithmic Functions. 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. > ; Domain: all (c) The graph is increasing on, and decreasing on,. (d) There are no relative maimum or relative minimum values.. f ln. f t log t, t f log f log. (c) f log. (d) ln ; Domain: (c) The graph is increasing on,. (d) There are no relative maimum or relative minimum values.. 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.
12 Section. Logarithmic Functions and Their Graphs.. ln. False. You would reflect in the line.. ln. ft min. log b. b b log b. The vertical asmptote is to the right of the -ais, and the graph increases. b Matches. b. f log is the inverse of g a a, where a >, a.. False, is not an eponential function of. ( can never be.) True, could be log. (c) True, could be. (d) False, is not linear. (The points are not collinear.). f ln (c). f..... As increases without bound, f approaches f g f g. fg f g.. The graphs of and The graphs of and intersect intersect when. or. when. or.
13 Chapter Eponential and Logarithmic Functions 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 log log ln ln ln ln. log a log log a log a ln ln a. log. log log. log. ln ln.. log ln. ln. log ln ln. log. ln. ln.. log ln. ln. ln ln. ln ln ln ln ln. ln ln ln ln log b log b log b... log b log b... f log ln ln. f log ln ln ln ln
14 Section. Properties of Logarithms. f log ln ln ln ln. 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. log log log. log log. ln z ln z ln z. ln z ln ln ln z. log a bc log a log b log c. ln a a ln a ln a log a log b log c ln a ln a, a >. ln ln. ln ln ln ln ln ln ln ln ln ln, > ln ln. ln z ln ln z ln ln ln z ln ln ln z. ln ln ln (c) The graphs and table suggest that for >. In fact, ln ln ln ln ln.
15 Chapter Eponential and Logarithmic Functions. ln ln ln. log z log log z. log 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 (c) The graphs and table suggest that. In fact, ln ln ln ln
16 Section. Properties of Logarithms. 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.. log log. log.. log. log.. log is undefined. is not in the domain of. log log log log log f log.. ln e ln e. ln e ln e. ln e ln ln e ln e. (c) log I log I log log I log log I log I I log log log log log log. T.. t T.. t CONTINUED The data t, T fits the model T.. t. The model T.. t fits the original data.
17 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.. True. False. For eample, let and a. Then f But f ln. a ln. f a ln. False. ln ln In fact, ln ln.. True. In fact, if ln <, then < <.. 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.. f log ln ln. f log ln ln. f log ln ln
18 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....,,..,, ±, ±. ±, ±
19 Chapter Eponential and Logarithmic Functions 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.. e e e e + e e No, e is not a solution. ln e ln e ln Yes, ln is a solution.. log.. is an approimate solution. No, is not a solution. (c) Yes, is a solution. (c). e. e. Yes,. is a solution.
20 Section. Solving Eponential and Logarithmic Equations. ln. e.. g ln e. ln e.. Yes, e. is a solution. f. Point of intersection:, ln. ln.. Algebraicall: Yes,. is a solution. (c) ln., ln ln. ln ln.. No, ln. is not a solution.. Point of intersection:, Algebraicall: f g,.. f g g f Point of intersection:, Algebraicall: log log, Point of intersection:, Algebraicall: ln e, Alternate solution: log log log.
21 Chapter Eponential and Logarithmic Functions. ln ln. ln. log. log ln ln e. ln. ln e ln e. e ln e e.. ln e.. t.. ln ln ln ln ln ln ln ln. t ln ln t ln ln t t ln ln ln ln ln ln... t.. t t ln. ln t t. ln ln...,... ln. ln. ln. ln... e. e ln ln. e. e ln ln ln. e e e ln
22 Section. Solving Eponential and Logarithmic Equations. e e. e.,. e e e e e. e or e. ln ln. e is impossible. ln..,. e e. e. e ± ±.,. e ln ln ln. e. e. e. e.. ln ln... e f..... e f.... t t.. e The zero of is.. e. g e. g t e.t. ln e. Zero at. Zero at t.
23 Chapter Eponential and Logarithmic Functions. ln.. ln. e. ln e. ln. e e. log log. log z. log.. ln z log. e z. e e... ln e ln e e e or e e. or e.. log log log log. ln ln ln ln ln or Both of these solutions are etraneous, so the equation has no solution.. log log ± log ± Choosing the positive value, we have. and..
24 Section. Solving Eponential and Logarithmic Equations. ln.. log. f..... f log. Graphing log, ou obtain two zeros,. and.. ln ln Graphing ln ln, ou obtain one zero,... ln ln. Graphing ln ln, ou obtain.. Intersection:.,.. e.. ln Intersection:., Intersection:.,.. e e. e e e e since e since e,. ln. ln ln since > ln e. ln ln ln e. e
25 Chapter Eponential and Logarithmic Functions. e.t. e.t e.t e.t ln.t ln.t t ln. ears. t ln. ears. e.t e.t e.t e.t ln.t ln.t t ln. ears. t ln. ears.. p. e... ln t p. ln t. e. ln t. e. t., or. ln units p. e. e.. ln units. f m From the graph we see horizontal asmptotes at and. These represent the lower and upper percent bounds. (c) Males: e.. e.... ln.. e... inches Females: e.. e.... ln.. e... inches
26 Section. Solving Eponential and Logarithmic Equations. T h (c) h h h We see a horizontal asmptote at. This represents the room temperature. h ln ln h ln h ln ln ln h h. hour. False. The equation e has no solutions.. Yes. The doubling time is given b P Pe rt e rt ln rt t ln r.. Answers will var. 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. f. f. f,, <
27 Chapter Eponential and Logarithmic Functions 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. This is an eponential growth model. Matches graph (c). log. This is a logarithmic model, and contains,. Matches graph. ln This is a logarithmic model. Matches graph (d).. Since A,e.t, the time to double is given b,,e.t e.t ln.t t ln. ears.. Amount after ears: A,e. $,.. Since A e rt and A, when t, we have the following., e r e r ln r r ln..% Amount after ears: A e. $,.
28 Section. Eponential and Logarithmic Models. Since A e rt and A. when. Since A Pe.t and A, when t, t, we have the following. 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.., Pe., e. P,. The time to double is given b,.,.e.t e.t ln.t t ln. ears... P Pe rt ln rt ln r e rt t r % % % % % % t ln r Continuous compounding results in faster growth. A. t and A e.t C Cek. ek k ln Ce kt e ln. g C Cek ek k ln Ce kt e ln. g
29 Chapter Eponential and Logarithmic Functions. ln b ln ae b. ae b a e b b b. Thus, e.., a, e b b ln ln. e.. Australia:,.,,. The constant b gives the growth rates. a. and..e b b. (c) The constant b is negative for South Africa..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. Turke:,.,,. a. and..e b b..e.t For,. million...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.%.
30 Section. Eponential and Logarithmic Models. V mt b, V, b, V,, m, V t t, V ae kt, V, b, V,,,e k V,e.t m k ln,., (c), (d) The eponential model depreciates faster in the first ear. (e) Answers will var.. S t e kt. e k e k k ln. k. S t e.t.e,. Maimum point is, the average IQ score. (c) S e.., units. 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. R log I I log I I R I.,, I.,, (c) I.,,,
31 Chapter Eponential and Logarithmic Functions. I log I I, where I watt per square meter. (c) log log decibels log log decibels log log decibels. log I I I I I % decrease I. I. I. I. ph log H log...%. ph log H ph log H ph H Hdrogen ion concentration of grape Hdrogen ion concentration of milk of magnesia... 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 u v u v when t. ears.. 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.. False. The domain could be all real numbers.. True. For the Gaussian model, >.
32 Section. Nonlinear Models.. Slope: Matches. Intercepts:,,,. Slope:. Matches (d). Intercepts:,,,. f. The graph falls to the left and rises to the right. g. The graph rises to the left and falls to the right... Answers will var. 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. Quadratic model. Eponential model. Quadratic model... Logarithmic model Eponential model Linear model
33 Chapter Eponential and Logarithmic Functions ln Coefficient of determination: Coefficient of determination: Coefficient of determination: ln... 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.. Linear model: P.t. Coefficient of determination:. Power model: P.t. Coefficient of determination:. CONTINUED
34 Section. Nonlinear Models. CONTINUED (c) Eponential model: P.. t Coefficient of determination:. (d) Quadratic model: P.t.t. Coefficient of determination:. (e) The quadratic model is best because its coefficient of determination is closest to. (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.. T.t. T.t.t. No, the data does not appear linear. Yes, the data appears quadratic. But, for t, the graph is increasing, which is incorrect. (c) Subtracting from the T-values, the eponential model is.. t. Adding back, T.. t. (d) Answers will var... P The model is a good fit..e.
35 Chapter Eponential and Logarithmic Functions. Linear model:.t. Logarithmic model:. ln t. Quadratic model:.t.t. Eponential model:.. t Power model:.t. 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.. True. Slope: -intercept:,.... Slope: -intercept:,
36 Review Eercises for Chapter Review Eercises for Chapter e.... e. e... f. f Intercept:, Intercept:, Horizontal asmptote: -ais Horizontal asmptote: -ais Increasing on:, Decreasing on:, Matches graph (c). Matches graph.. f. g. Intercept:, Horizontal asmptote: -ais Increasing on:, Intercept:, Horizontal asmptote: Decreasing on:, h e Horizontal asmptote:. h e.. f e. f e. Horizontal asmptotes:, Horizontal asmptote: Horizontal asmptote:. A Pe rt,e.t t A,.,.,.,.,.,.
37 Chapter Eponential and Logarithmic Functions. V t, t, For t, V $,. (c) The car depreciates most rapidl at the beginning, which is realistic.. log. log. ln e e e. g lo.. log log log. e..... log log. log log ln.... log log. g log ln ln. f log ln ln Domain: > Vertical asmptote: -intercept:, Domain: > Vertical asmptote: -intercept:.,. ln... ln.. log log. log log. f ln. h ln Domain:, Vertical asmptote: -intercept:., Domain: > 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,
38 Review Eercises for Chapter. log log.. log log ln. ln log log log log ln ln... f log ln ln. ln ln f log ln ln. log b log b. log b log b. ln e ln ln e log b log b ln ln e.. ln.... log log. log log log log log log log log log log. log log log ln ln ln ln ln ln log log log. log log log. 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.
39 Chapter Eponential and Logarithmic Functions.... log. log. ln. ln. e e. e e e ln ln... ln ln ln ln. ln ln ln ln.. e. e e e e e e ln. e ln ln. e ln.. ln.. ln ln. ln ln e. ln e.. ln e e. ln ln e e.
40 Review Eercises for Chapter. log log log. log log No solution (etraneous) log.. e e. ln e ln since e ln since > ln e. e. e.t. e e.t Decreasing eponential ln.t Matches graph (e). t ln.. ears. ln. Logarithmic function shifted to left Matches graph (f). e Gaussian model Matches graph.. ae b. ae b a e b. e b ln. b b. Thus, e... ae b ae b a eb e b ln b b. Thus, e..
41 Chapter Eponential and Logarithmic Functions. P e kt t corresponds to., :.,,e r e r ln r e k e k r ln. or.%,e. $,. k ln k ln ln. P e.t For, P e.. or, population in...e.t.e.t e.t..t ln.e.t Similarl: t ln.. weeks.e.t.e.t e.t..t ln. t. weeks. Logistic model. Logarithmic model. Linear model:.t ;. Quadratic model:.t.t ;. Eponential model:.. t ;. Logarithmic model:. ln t ;. Power model:.t. ;. CONTINUED
42 Review Eercises for Chapter. CONTINUED 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.. P..e.. True; b the Inverse Properties, log b b., (c) The model is a good fit. (d) The limiting size is., fish.. False; ln ln ln ln. False. The domain of f ln is >.. Since < <, < < < <.
43 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. Houghton Mifflin Compan. All rights reserved.. Use a graphing utilit to find the power model a b for the data,,,,,, and,.
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