Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 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 ab ; a, b, b a: initial value (-intercept); f ab a b: base 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 before 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 Substitute a and,, a b to find b., and,. b b b Note: As stated above, b, so we will use onl b. Writing an Eponential Function Given Two Points E: Find an eponential function that contains the points Substitute each point into a b for and. Divide the two equations to eliminate a. Substitute this value for b to solve for a. a b 6 ab ab, and,6. 6 a b b a a Solution: Note: Another method for solving the sstem of equations above is to solve for a and use the substitution method. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Writing an Eponential Function Given a Table of Values E: Determine a formula for the table of values. 6 7 7/ Check for a common ratio (b): b Initial Value (a): value of when : a 7 Solution: 7, 6 7, 7/ 7 Note: If the table did not give the initial value, then we can use one of the points given for and, and b in the equation a b to solve for a. Challenge Problem: 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 base, 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 Possible Solution: f f f m f n f f 8 f 8 Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Graphs of Eponential Functions E: Plot points and graph the functions and. /9 / 9 9 / /9 Eponential Growth Eponential Deca Note: Each of the graphs above have a horizontal asmptote at. Graphs of Eponential Functions: Graph is INCREASING when b& a ; graph is DECREASING when b& a. Transformations of Eponential Functions: h ab 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: Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Eponential Model: ab Eponential Growth: a & b ; b is the growth factor Eponential Deca: a & b ; b 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 number of students. Substitute the given values into the eponential model ab and solve for a and b. 6 a b 9 ab 6 ab 9 a b 9 9.77 b.77 b b.6 (Eponential Growth) a 6 a.6 7.7 7.7.6 Population in : 7.7.6 6 87,7students nt r Compound Interest: A P n A = balance amount P = principal (beginning) amount r = annual interest rate (decimal) n = # of times compounded in a ear t = time in ears Note: Annuall = time per ear Semiannuall = times per ear Quarterl = times per ear Monthl = times per ear Weekl = times per ear r A P n E: Calculate the balance if $ 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) be equal to some constant (m) multiplied b the rate: n mr. r A P P P mr m m mrt mrt m rt m m Graph and find lim. Make a table. m m m The values of are approaching.78. This is the approimate value of the transcendental number e. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions 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 b The logistic function is used for populations that will be limited in their abilit to grow due to limited resources or space. Think About It: What would limit population growth? 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:, Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions E: Sketch the graph of the function. State the -intercept and horizontal asmptotes.. H.A.:,. -intercept: You Tr: Describe the transformations needed to draw the graph of f. Sketch the graph. QOD: Using a table of values, how can ou determine whether the have an eponential relationship? Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objectives: 9.7 The student will solve application problems involving eponential and logarithmic functions. 9. The student will solve eponential, logarithmic and logistic equations and inequalities. t Eponential Model: P t Pb P t : population at time t P : initial population Growth Model: b r; b is called the Growth Factor Deca Model: b r; b is called the Deca Factor E: Write an eponential function that models the population of Smallville if the initial population was,8, and it is decreasing b.% each ear. Predict how long it will take for the population to fall to. P 8 r., so b..977 P t 8.977 t Solve for t when P t. 8.977 t It will take about.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 Pb and write a sstem of equations with the given information: Pb Pb Pb Solve the sstem b dividing the equations: Pb P P P t P t P 9 ants Note: Students could have called da t to come up with the same solution. b b.7 Pb Eponential Regression E: Find an eponential regression for the population of Las Vegas using the table below. Then predict the population in 9. 9 9 97 99 8,,6,787 8,9,87 9: 896,767 Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Radioactive Deca: the process in which the number 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 substance 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 Pb : Solve for b when P, P., and t 8. 8. b b. 8 b.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,, b? a b 9 b b9 b b f 9 You Tr: Complete the table. 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 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of an eponential, logistic or logarithmic function. 9. The student will evaluate eponential or logarithmic epressions. 9. The student will graph the inverse of an eponential or logarithmic function. 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 above) Now use our calculator to find LOG 997. Common Logarithm: Given a positive number p, the solution to p is called the base- logarithm of p, epressed as log p, or simpl as log p. (When no base is specified, it is understood to be base.) A LOGARITHM IS AN EXPONENT. Logarithm (base b): log b (Read as log base b of. ) for, b and b if and onl if b. Note: You cannot take the log of a negative number! E: Rewrite each equation (eponential form) to logarithmic form... Base (b) = Eponent = log or log 6 Base (b) = Eponent = log6 E: Rewrite each equation (logarithmic form) to eponential form.. log Base (b) = Eponent =. log, Base (b) = Eponent =, Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Evaluating Logarithms: a logarithm is an eponent 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 table 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 table, 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 b k alog b h Vertical Stretch: a, a Horizontal Stretch: a, a Reflection over -ais: a Reflection over -ais: Horizontal Translation: h Vertical Translation: k : Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions log. Then E: Describe the transformations used to graph the function sketch the graph. Reflect over -ais; shift left units; shift down units Note: Graph in red is the graph of log. Properties of Logarithms: loga because a log a a because a a log a because loga a a a or a (propert of inverses) If log log, then a a Natural Logarithmic Function: logarithm base e, log e, written ln Notations:. log is used to represent log (common logarithm). ln is used to represent log e (natural logarithm) 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 above eamples without rewriting using the following: log and log a a a a Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Solving Logarithmic Equations: E: Solve the equations for.. log7 log76 The bases are equal, so 6 6. Note: Both solutions work in the original equation.. log9 Rewrite eponentiall: 9. ln e Rewrite eponentiall: e e Note: Remember that the base of ln is 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 number? Eplain. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will appl the properties of logarithms to evaluate epressions, change bases, 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 b b b Because a logarithm is an eponent, the rules are the same! b b b b b Properties of Logarithms: log RS log R log S b b b log R log R log S c log R c log R b b b b b 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 Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions E: Use the properties of logarithms to condense the following epressions.. log log log log log. log log. lnlnln log log log log log lnlnlnln lnln ln ln Evaluating Logarithmic Epressions with Base b Let log b. Rewrite in eponential form: b. Take the log of both sides. log b log log Use the properties of logs to solve for. log blog log b Note: This will work for a logarithm of an base, including the natural log. Change of Base Formula: log a ln a logb a log b ln b E: Evaluate the epression log 8. log8 B the change of base formula, log 8. Using the calculator, log8.9 log log E: Evaluate the epression log6. ln B the change of base 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 Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions 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 base into the calculator, so we must rewrite the functions using the change of base formula. log log log log log log Caution: The graph created b 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 base formula? Eplain how to evaluate a logarithm of base b without the change of base formula. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objectives: 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 both sides with the same base Take the log of both sides after isolating the eponential E: Solve the following eponential equations.. Rewrite with base : Both sides have the same base, so the eponents must be equal:. 8 6 We cannot rewrite both sides with the same base, so take the log of both 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 both 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 base 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 both 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 taking the th root of each side. 6, b Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions. log log 8 Condense: Rewrite in eponential form: 8 log 8 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 possible 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 Problems: 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 both sides: ln ln 7 Rewrite: ln ln7 Solve for : ln ln7ln7 ln ln7 ln7 lnln7 ln7 ln 7.879 Check ln ln7 Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions You Tr: Solve the equation. e QOD: Compare and contrast the methods for solving eponential and logarithmic equations. Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9.7 The student will solve application problems involving eponential and logarithmic functions. T t T T T e kt s S, where Newton s Law of Cooling: The temperature T of an object at time t is T s is the surrounding temperature and T is the initial temperature of the object. 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 about 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 = balance r = annual interest rate n P = principal t = time in ears n = number 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 about. ears. ln. Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter
Precalculus Notes: Unit Eponential and Logarithmic Functions E: How long will it take for an investment of $, at 6% compounded continuousl to reach $? ln rt.6t.6t A Pe e ln ln e.6t ln t.7.6 It will take about.7 ears. 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 be invested at 9% interest compounded monthl to produce a balance of $, in ears. QOD: Wh is using the annual percentage ield a more fair wa to compare investments? Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter