Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic epressions. Eponential Function: a function of the form f a a,, a: initial value (-intercept); f a a : ase Evaluating an Eponential Function E: Determine the value for f. Note: A common mistake is multipling the and. Remind students of the order of operations (eponents efore multiplication). Or have students write the function as f.. f f 6 9. f f 6. f f 6 Writing an Eponential Function Given the Initial Value E: Find an eponential function that contains the points Initial Value: a Sustitute a and,, a to find., and,. Note: As stated aove,, so we will use onl. Writing an Eponential Function Given Two Points E: Find an eponential function that contains the points Sustitute each point into a for and. Divide the two equations to eliminate a. Sustitute this value for to solve for a. a 6 a a, and,6. 6 a a a Solution: Writing an Eponential Function Given a Tale of Values Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions E: Determine a formula for the tale of values. 6 7 7/ Check for a common ratio (): Initial Value (a): value of when : a 7 Solution: 7, 6 7, 7/ 7 Challenge Prolem: Find f such that f mn f m f n for all m and n. Students should recall that when multipling eponentials with the same ase, the eponents are added. So it should make sense to hpothesize that an eponential function would work for f. Tr a specific eample: f, with m, n. f mn f f f Possile Solution: f f f m f n f f 8 f 8 Graphs of Eponential Functions E: Plot points and graph the functions and. /9 / 9 9 / /9 Note: Each of the graphs aove have a horizontal asmptote at. Graphs of Eponential Functions: Graph is INCREASING when & a ; graph is DECREASING when & a. Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Transformations of Eponential Functions: h a k Vertical Stretch: a Vertical Shrink: a Reflection over -ais: a Reflection over -ais: Horizontal Translation: h Vertical Translation: k Horizontal Asmptote: k E: Graph the eponential function. Transformation of. Rewrite as. Horizontal Translation: right Reflect over -ais. Vertical Translation: up -intercept: f Horizontal Asmptote: Eponential Model: a Eponential Growth: a & ; is the growth factor Eponential Deca: a & ; is the deca factor Applications of the Eponential Model E: CCSD s student population went from, in 96 to 9, students in. Write an eponential function that represents the student population. Predict the population in. Let represent the ear 9 and represent the numer of students. Sustitute the given values into the eponential model a and solve for a and. 6 a 9 a 6 a 9 a 9 9.77.77.6 a 6 a.6 7.7 7.7.6 Population in : 7.7.6 6 87,7students nt r Compound Interest: A P A = alance amount P = principal (eginning) amount n r = annual interest rate (decimal) n = # of times compounded in a ear t = time in ears Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions r A P n E: Calculate the alance in $ is invested for ears at 6% compounded weekl. nt A? P t r.6 n.6 A $6.7 Isolating the change in the compounding period reveals a naturall occurring constant. Let the compounding period (n) e equal to some constant (m) multiplied the rate: n mr. r A P P P mr m m mrt mrt m rt m m Graph and find lim. Make a tale. m m m The values of are approaching.78. This is the approimate value of the transcendental numer e. Natural Base e: e lim.7888 Interest Compounded Continuousl: A Pe rt k Natural Eponential Function: f a e Graph of e : c Logistic Function: f c (constant): limit to growth (maimum) a The logistic function is used for populations that will e limited in their ailit to grow due to limited resources or space. Think Aout It: What would limit population growth? Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions E: Estimate the maimum population for Dallas and find the population for the ear 8 given,,6 the function Pt () that models the population from 9..t.6e Maimum population: c,,6 Population in 8: t 8 Graph of a Logistic Function: f Domain: All Reals Range:,,6 P(8).8,9,96.6e e, Alwas Increasing Horizontal Asmptotes:, E: Sketch the graph of the function. State the -intercept and horizontal asmptotes.. H.A.:,. -intercept: You Tr: Descrie the transformations needed to draw the graph of f. Sketch the graph. QOD: Using a tale of values, how can ou determine whether the have an eponential relationship? Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojectives: 9.7 The student will solve application prolems involving eponential and logarithmic functions. 9. The student will solve eponential, logarithmic and logistic equations and inequalities. P t : population at time t P : initial population Growth Model: r; is called the Growth Factor Deca Model: r; is called the Deca Factor t Eponential Model: Pt P E: Write an eponential function that models the population of Smallville if the initial population was,8, and it is decreasing.% each ear. Predict how long it will take for the population to fall to. P 8 r., so..977 P t 8.977 t Solve for t when P t. 8.977 t It will take aout.66 ears. E: The population of ants is increasing eponentiall such that on da there are ants, and on da there are ants. How man ants are there on da? t Use P t P and write a sstem of equations with the given information: P P P Solve the sstem dividing the equations: P P P P t P t P 9 ants Note: Students could have called da t to come up with the same solution..7 P Eponential Regression E: Find an eponential regression for the population of Las Vegas using the tale elow. Then predict the population this ear. 9 9 97 99 8,,6,787 8,9,87 9: 896,767 Page 6 of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Radioactive Deca: the process in which the numer of atoms of a specific element change from a radioactive state to a nonradioactive state Half-Life: the time it takes for half of a sample of a radioactive sustance to change its state E: The half-life of radioactive Strontium is 8 das. Write an equation and predict the amount of a gram sample that remains after das. t Use P t P : Solve for when P, P., and t 8. 8.. 8.97 Equation: P t P.97 t -gram sample remaining after da: P t.97.8 grams Writing a Logistic Function f E: Find a logistic function that satisfies the given conditions: Initial value = ; limit to growth = ; passes through the point,. c c, a,,? a 9 9 f 9 You Tr: Complete the tale. Isotope Half-Life (ears) Initial Quantit Amount After Years C 7 grams C 7.7 gram 6 Ra 6 grams QOD: Eplain how to determine if an eponential function is a growth or deca model. Page 7 of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of an eponential, logistic or logarithmic function. 9. The student will evaluate eponential or logarithmic epressions. Review: Solve the following for...... Eploration: Use our calculator to write 997 as a power of. Note:,... Tr 8.9, 8.9, 99. (close!) log 997.78 (Compare to aove) Now use our calculator to find LOG 997. Common Logarithm: Given a positive numer p, the solution to of p, epressed as log p, or simpl as log p. A LOGARITHM IS AN EXPONENT. p is called the ase- logarithm Logarithm (ase ):. log (Read as log ase of. ) for, and if and onl if E: Rewrite each equation (eponential form) to logarithmic form... Base () = Eponent = log or log 6 Base () = Eponent = log6 E: Rewrite each equation (logarithmic form) to eponential form.. log Base () = Eponent =. log, Base () = Eponent =, Evaluating Logarithms: a logarithm is an eponent Page 8 of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions E: Evaluate the logarithms.. log 8 8. f, f log log 6 f 6 6. log 6. f., f log f log. Calculator: f. Graphing Logarithmic Functions: a logarithmic function is the inverse of an eponential function E: Use a tale of values to sketch the graph of log. Discuss the characteristics of the graph and compare the graph to the graph of. Note: To create the tale, it is helpful to rewrite the function as and choose values for. 9 / /9 Domain: Increasing:, Range:, Intercepts: int :,, End Behavior: lim f lim f Asmptote: The function log is the inverse of, so its graph is the reflection of over the line. Transformations of log : k a h log Vertical Stretch: a, a Horizontal Stretch: a, a Reflection over -ais: a Reflection over -ais: Horizontal Translation: h Vertical Translation: k E: Descrie the transformations used to graph the function sketch the graph. Reflect over -ais; shift left units; shift down units log. Then Note: Graph in red is the graph of log. Page 9 of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Properties of Logarithms: loga ecause a log a a ecause a a log a ecause loga a a a or a (propert of inverses) If log log, then a a Natural Logarithmic Function: logarithm ase e, written ln Evaluating Logarithmic Epressions:. E: Evaluate log. log. Let log log. Let and log... Rewrite in eponential form:.. log.. Rewrite in logarithmic form: log log.. Note: B the properties of inverses, we could have evaluated the aove eamples without rewriting using the following: log and log a a a a Solving Logarithmic Equations:. E: Solve the equations for. log7 log76 The ases are equal, so 6 6. Note: Both solutions work in the original equation.. log9 Rewrite eponentiall:. ln e Rewrite eponentiall: Note: Rememer that the ase of ln is e. 9 e e You Tr: Sketch the graph of function on the same grid with its inverse. f ln QOD: Can ou evaluate the log of a negative numer? Eplain. Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will appl the properties of logarithms to evaluate epressions, change ases, and re-epress data. Eploration: Use our calculator to find log and log. Evaluate the following logarithms on our calculator, then speculate how ou could calculate them using onl the values of log and/or log. log. and log.77. log6 log6.778.778..77 log log. log8 log8.9.9. log. log log.76.76..77 log log Recall: Properties of Eponents Because a logarithm is an eponent, the rules are the same! Properties of Logarithms: log RS log R log S log R log R log S c log R c log R S E: Use the properties of logarithms to epand the following epressions.. log. ln z. log. log log log log ln ln ln ln z z log log log log log log log log log log log log log log E: Use the properties of logarithms to condense the following epressions.. log log log log log Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions log log log log log. log log. lnlnln lnlnlnln lnln ln ln Evaluating Logarithmic Epressions with Base Let log. Rewrite in eponential form:. Take the log of oth sides. log log log Use the properties of logs to solve for. log log log Note: This will work for a logarithm of an ase, including the natural log. Change of Base Formula: log a ln a log a log ln E: Evaluate the epression log 8. log8 B the change of ase formula, log 8. Using the calculator, log8.9 log log E: Evaluate the epression log6. ln B the change of ase formula, log6. Using the calculator, ln.67 ln 6 ln 6 Note This did NOT require the use of a calculator! We know that 6 6. So log6 6 Graphing Logarithmic Functions on the Calculator E: Graph log and log on the same grid on the graphing calculator. We cannot tpe in a log ase into the calculator, so we must rewrite the functions using the change of ase formula. log log log log log log Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Caution: The graph created the calculator is misleading at the asmptote! You Tr: Epand the epression using the properties of logarithms. ln. z QOD: When is it appropriate to use the change of ase formula? Eplain how to evaluate a logarithm of ase without the change of ase formula. Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojectives: 9. The student will solve eponential, logarithmic and logistic equations and inequalities. 9.6 The student will compare equivalent logarithmic and eponential equations. Strategies for Solving an Eponential Equation: Rewrite oth sides with the same ase Take the log of oth sides after isolating the eponential E: Solve the following eponential equations.. Rewrite with ase : Both sides have the same ase, so the eponents must e equal:. 8 6 We cannot rewrite oth sides with the same ase, so take the log of oth sides. log 6 log8 log 6 log8 log 6 Solve for :.9 log8. 7e Isolate the eponential: e e ln Take the natural log of oth sides: ln e ln ln eln ln e Note: We could have determined that immediatel using the equation e. Strategies for Solving a Logarithmic Equation: Condense an logarithms with the same ase using the properties of logs Rewrite the equation in eponential form Check for etraneous solutions E: Solve the following logarithmic equations.. ln ln Condense: ln Rewrite in eponential form: e Solve for : Check: ln ln. log 6 Rewrite in ep. form: 6 ln 6 Take the log of oth sides: ln ln 6 ln ln 6 ln ln 6 Rewrite in eponential form: e Note: We could have determined that immediatel using the equation 6.. loglog8 Condense: Rewrite in eponential form: 8 log 8 Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions 9 8 8 9 9, 9 Solve for : Check: : log log 8 (can t take the log of a negative!) 9 : log 9 log 9 8 log 9 Note: You must check ever possile solution for etraneous solutions. All negative answers are not necessaril etraneous! E: Solve the equation log. Rewrite in eponential form: Solve for : log log Check: E: Solve the equation log 6. Rewrite using properties of logs: log 6 log Rewrite in eponential form: Check: 6 log log log 6 Challenge Prolems: Use our arsenal of eponential and logarithmic properties! E: Solve the equation 6 7. Rewrite the first term: Solve for : 6 7 Let 67. 7,7 Use to solve for : : (no solution) 7 : log7 7 log log7 log log7.87 log E: Solve the equation 7. Take the log of oth sides: ln ln 7 Rewrite: ln ln7 ln ln7ln7 ln ln7 ln7 lnln7 ln7 ln 7.879 Check ln ln7 Solve for : You Tr: Solve the equation. e QOD: Compare and contrast the methods for solving eponential and logarithmic equations. Page of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9.7 The student will solve application prolems involving eponential and logarithmic functions. kt s S, where T t T T T e Newton s Law of Cooling: The temperature T of an oject at time t is T s is the surrounding temperature and T is the initial temperature of the oject. E: A F potato is left out in a 7 F room for minutes, and its temperature dropped to F. How man more minutes will it take to reach F? kt T t T T T e Solve for k using the given information: k k 9 k 7 7e 8 8e e 9 ln 9 k 9 ln ln ln e k k.68 Use k to solve for t:.68t.68t 7 7e e ln.68t t 6.8 8 8 It takes aout 6.8 minutes for the potato to cool to F. This is 6.8 =.8 minutes longer. r Formula for Compound Interest: A P A = alance r = annual interest rate n P = principal t = time in ears n = numer of times interest is compounded each ear nt s S Interest Compounded Continuousl: A Pe rt E: How long will it take for an investment of $, at 6% compounded semi-annuall to reach $? r A P n nt t.6 t t. ln ln. ln ln. t ln t.99 It will take aout. ears. ln. E: How long will it take for an investment of $, at 6% compounded continuousl to reach $? Page 6 of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions rt ln A Pe e e t t.6 It will take aout.7 ears..6t.6t ln ln.6 ln.7 Annual Percentage Yield (APY): the rate, compounded annuall ( t ), that would ield the same return nt n r For A P n, APY r n E: An amount of $ is invested for 8 ears at % compounded quarterl. What is the equivalent APY? n r. APY.9.9% n You Tr: Determine the amount of mone that should e invested at 9% interest compounded monthl to produce a alance of $, in ears. QOD: Wh is using the annual percentage ield a more fair wa to compare investments? Page 7 of 7 Precalculus Graphical, Numerical, Algeraic: Pearson Chapter