Exponential and Logarithmic Functions

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1 Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define ech term or concept. Algebric functions Trnscendentl functions Nturl bse e Continuous compounding I. Eponentil Functions (Pge 28) The eponentil function f with bse is denoted b, where > 0,, nd is n rel number. How to recognize nd evlute eponentil functions with bse Emple : Use clcultor to evlute the epression / 5 5. II. Grphs of Eponentil Functions (Pges 29 22) For >, is the grph of its domin? = For >, is the grph of = incresing or decresing over incresing or decresing over How to grph eponentil functions nd use the One-to-One Propert its domin? For the grph of = or =, >, the domin is, the rnge is, nd the intercept is. Also, both grphs hve s horizontl smptote. Copright Houghton Mifflin Compn. All rights reserved. 57

2 58 Chpter Eponentil nd Logrithmic Functions Emple 2: Sketch the grph of the function f ( ) = The grph of the eponentil function psses the Test, nd therefore, the function is one-to-one function (nd, thus, hs n inverse function). Stte the One-to-One Propert for eponentil functions nd eplin how it m be used to solve simple eponentil equtions. III. The Nturl Bse e (Pge 222) The nturl eponentil function is given b the function. In this function, is the constnt nd is the vrible. How to recognize, evlute, nd grph eponentil functions with bse e / 5 Emple : Use clcultor to evlute the epression e. Copright Houghton Mifflin Compn. All rights reserved.

3 Section. Eponentil Functions nd Their Grphs 59 IV. Applictions of Eponentil Functions (Pges ) After t ers, the blnce A in n ccount with principl P nd nnul interest rte r (in deciml form) is given b the formuls: For n compoundings per er: How to use eponentil functions to model nd solve rel-life pplictions For continuous compounding: Emple 4: Find the mount in n ccount fter 0 ers if $6000 is invested t n interest rte of 7%, () compounded monthl. (b) compounded continuousl. Additionl notes Copright Houghton Mifflin Compn. All rights reserved.

4 60 Chpter Eponentil nd Logrithmic Functions Additionl notes Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.

5 Section.2 Logrithmic Functions nd Their Grphs 6 Nme Section.2 Logrithmic Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph logrithmic functions. Importnt Vocbulr Define ech term or concept. Common logrithmic function Nturl logrithmic function I. Logrithmic Functions (Pges 229 2) For > 0, > 0, nd, where =, the logrithmic function with bse is defined s., which is red s How to recognize nd evlute logrithmic functions with bse The logrithmic function with bse is the of the eponentil function f ( ) =. The eqution = in eponentil form is equivlent to the eqution in logrithmic form. When evluting logrithms, remember tht logrithm is (n). This mens tht log is the to which must be rised to obtin. Emple : Use the definition of logrithmic function to evlute log Emple 2: Use clcultor to evlute log Copright Houghton Mifflin Compn. All rights reserved.

6 62 Chpter Eponentil nd Logrithmic Functions Complete the following properties of logrithms: ) log = 2) log = ) log = nd log = 4) If log = log, then. Emple : Solve the eqution log 7 = for. II. Grphs of Logrithmic Functions (Pges 2 22) To sketch the grph of = log, ou cn use the fct tht... How to grph logrithmic functions For >, is the grph of over its domin? = log incresing or decresing For the grph of = log, >, the domin is, the rnge is, nd the -intercept is. Also, the grph hs s verticl smptote. Emple 4: Sketch the grph of the function ( ) = log. f Copright Houghton Mifflin Compn. All rights reserved.

7 Section.2 Logrithmic Functions nd Their Grphs 6 Nme III. The Nturl Logrithmic Function (Pges 2 24) The nturl logrithm is written without bse; the bse is understood to be. How to recognize, evlute, nd grph nturl logrithmic function Complete the following properties of nturl logrithms: ) ln = 2) ln e = ) ln e = nd e ln = 4) If ln = ln, then. Emple 5: Use clcultor to evlute ln 0. Emple 6: Find the domin of the function f ( ) = ln( + ). IV. Applictions of Logrithmic Functions (Pge 25) Describe rel-life sitution in which logrithms re used. How to use logrithmic functions to model nd solve rel-life pplictions Emple 7: A principl P, invested t 6% interest nd compounded continuousl, increses to n mount K times the originl principl fter t ers, where t ln K is given b t =. How long will it tke the 0.06 originl investment to double in vlue? To triple in vlue? Copright Houghton Mifflin Compn. All rights reserved.

8 64 Chpter Eponentil nd Logrithmic Functions Additionl notes Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.

9 Section. Properties of Logrithms 65 Nme Section. Properties of Logrithms Objective: In this lesson ou lerned how to use the chnge-of-bse formul to rewrite nd evlute logrithmic epressions nd how to use properties of logrithms to evlute, rewrite, epnd, or condense logrithmic epressions. I. Chnge of Bse (Pge 29) Let, b, nd be positive rel numbers such tht nd b. Use the Chnge-of-Bse Formul to rewrite log using bse b: log = How to use the chngeof-bse formul to rewrite nd evlute logrithmic epressions Eplin how to use clcultor to evlute log II. Properties of Logrithms (Pge 240) Let be positive number such tht ; let n be rel number; nd let u nd v be positive rel numbers. Complete the following properties of logrithms:. log ( uv) = How to use properties of logrithms to evlute or rewrite logrithmic epressions 2. u log = v n u. log = III. Rewriting Logrithmic Epressions (Pge 24) To epnd logrithmic epression mens to.... How to use properties of logrithms to epnd or condense logrithmic epressions Emple : Epnd the logrithmic epression 4 ln. 2 Copright Houghton Mifflin Compn. All rights reserved.

10 66 Chpter Eponentil nd Logrithmic Functions To condense logrithmic epression mens to.... Emple 2: Condense the logrithmic epression log + 4 log( ). IV. Applictions of Properties of Logrithms (Pge 242) One w of finding model for set of nonliner dt is to tke the nturl logrithm of ech of the -vlues nd -vlues of the dt set. If the points re grphed nd fll on stright line, then the -vlues nd the -vlues re relted b the eqution: How to use logrithmic functions to model nd solve rel-life pplictions stright line., where m is the slope of the Emple : Find nturl logrithmic eqution for the following dt tht epresses s function of Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.

11 Section.4 Eponentil nd Logrithmic Equtions 67 Nme Section.4 Eponentil nd Logrithmic Equtions Objective: In this lesson ou lerned how to solve eponentil nd logrithmic equtions. I. Introduction (Pge 246) Stte the One-to-One Propert for eponentil equtions. How to solve simple eponentil nd logrithmic equtions Stte the One-to-One Propert for logrithmic equtions. Stte the Inverse Properties for eponentil equtions nd for logrithmic equtions. Describe how the One-to-One Properties nd the Inverse Properties cn be used to solve eponentil nd logrithmic equtions. Emple : () Solve log 8 = for. (b) Solve 5 = 0.04 for. II. Solving Eponentil Equtions (Pges ) Describe how to solve the eponentil eqution 0 = 90. How to solve more complicted eponentil equtions Copright Houghton Mifflin Compn. All rights reserved.

12 68 Chpter Eponentil nd Logrithmic Functions 2 Emple 2: Solve e plces. 7 = 59 for. Round to three deciml III. Solving Logrithmic Equtions (Pges ) Describe how to solve the logrithmic eqution log6 (4 7) = log6 (8 ). How to solve more complicted logrithmic equtions Emple : Solve 4 ln 5 = 28 plces. for. Round to three deciml IV. Applictions of Eponentil nd Logrithmic Equtions (Pges ) Emple 4: Use the formul for continuous compounding, rt A = Pe, to find how long it will tke $500 to triple in vlue if it is invested t 2% interest, compounded continuousl. How to use eponentil nd logrithmic equtions to model nd solve rellife pplictions Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.

13 Section.5 Eponentil nd Logrithmic Models 69 Nme Section.5 Eponentil nd Logrithmic Models Objective: In this lesson ou lerned how to use eponentil growth models, eponentil dec models, Gussin models, logistic growth models, nd logrithmic models to solve rel-life problems. Importnt Vocbulr Define ech term or concept. Bell-shped curve Logistic curve Sigmoidl curve I. Introduction (Pge 257) The eponentil growth model is The eponentil dec model is. How to recognize the five most common tpes of models involving eponentil nd logrithmic functions The Gussin model is. The logistic growth model is Logrithmic models re nd II. Eponentil Growth nd Dec (Pges ) Emple : Suppose popultion is growing ccording to the 0.05t model P = 800e, where t is given in ers. () Wht is the initil size of the popultion? (b) How long will it tke this popultion to double? How to use eponentil growth nd dec functions to model nd solve rel-life problems Copright Houghton Mifflin Compn. All rights reserved.

14 70 Chpter Eponentil nd Logrithmic Functions To estimte the ge of ded orgnic mtter, scientists use the crbon dting model, which denotes the rtio R of crbon 4 to crbon 2 present t n time t (in ers). Emple 2: The rtio of crbon 4 to crbon 2 in fossil is R = 0 6. Find the ge of the fossil. III. Gussin Models (Pge 26) The Gussin model is commonl used in probbilit nd sttistics to represent popultions tht re. How to use Gussin functions to model nd solve rel-life problems For Gussin model, the verge vlue for popultion cn be found... Emple : Drw the bsic form of the grph of Gussin model. Copright Houghton Mifflin Compn. All rights reserved.

15 Section.5 Eponentil nd Logrithmic Models 7 Nme IV. Logistic Growth Models (Pge 262) Give n emple of rel-life sitution tht is modeled b logistic growth model. How to use logistic growth functions to model nd solve rel-life problems Emple 4: Drw the bsic form of the grph of logistic growth model. V. Logrithmic Models (Pge 26) Emple 5: The number of kitchen widgets (in millions) demnded ech er is given b the model = 2 + ln( + ), where = 0 represents the er 2000 nd 0. Find the er in which the number of kitchen widgets demnded will be 8.6 million. How to use logrithmic functions to model nd solve rel-life problems Copright Houghton Mifflin Compn. All rights reserved.

16 72 Chpter Eponentil nd Logrithmic Functions Additionl notes Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.

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