Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an Eponential Function E: Create a tale of values to graph the eponential function. 0 8 4 4 8 0 End Behavior: As, f. As, f 0, therefore 0 is an asmptote. - Eplore: Graph the following eponential functions and descrie the change from the graph of.. General characteristics of h a k :. Graph of a is shifted horizontall h units.. Graph of 4 a is shifted verticall k units. 4. 4 If a > 0 and >, it is an eponential growth function.. Domain of an Eponential Growth Function: All Reals 6. 4 Range of an Eponential Growth Function: k 7. E: Sketch the graph of 4 0 Step One: Sketch the graph of - Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions 0 Step Two: Sketch the graph of reflecting over the -ais. - Step Three: Translate the graph verticall 4 units up. 0 - -0 - - -0 Eponential Growth Models: In a real-life situation, if a quantit increases r percent each time period t, the situation can e modeled the equation a r r is called the growth factor. t, where a is the initial amount. The quantit E: In 990 the cost of tuition at a state universit was $400. During the net 8 ears, the tuition rose 4% each ear. Write a model that gives the tuition (in dollars) t ears after 990. Then estimate the cost of tuition in 999. Model: t 400 0.04 400.04 t 999: t 9 9 400.04 $60.4 nt r Compound Interest: A P A = amount in account after t ears, P = principal (amount n when t = 0, r = annual interest rate, n = numer of times per ear the interest is compounded E: Jane deposits $00 in an account that pas 6% annual interest. Find the alance after ears if the interest is compounded semiannuall. P = 00, r = 0.06, t =, n = 0.06 A 00 $79.08 You Tr:. Sketch the graph of the eponential function.. Use the eample aove and find out the alance of Jane s account if the interest is compounded quarterl. QOD: What is the difference etween percent increase and growth factor? Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sample CCSD Common Eam Practice Question(s): What function descries the graph elow? A. B. C. D. 6 Sample SAT Question(s): Taken from College Board online practice prolems.. If is a positive integer, what is one possile value of the units digit of 0 after it has een multiplied out? Grid-In Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Deca Function: f a, where a > 0 and 0 < < Domain: All Real Numers; Range: > 0 E: Graph the function and state its domain and range. 4 intercept: 0, Asmptote: 0 0 Domain: All real numers Range: 0 - E: State whether f is an eponential growth or deca function.. f. f 4 4 DECAY (ecause a 0 and ) GROWTH (ecause a 0 and ) 6 4.) 8 GROWTH (ecause a 0 and ) f DECAY (ecause a 0 and ). f 64 DECAY (Can e rewritten as f 4. f. Eponential Deca Models: In a real-life situation, if a quantit decreases r percent each time period t, t a r, where a is the initial amount. The quantit the situation can e modeled the equation r is called the deca factor. E: There are 40,000 homes in a certain cit. Each ear 0% of the homes are epected to disconnect from septic sstems and connect to the sewer sstem. Write an eponential deca model for the numer of homes that still use septic sstems. Use the graph of the model to estimate when aout 7,00 homes will still not e connected to the sewer sstem. Model: 40000 0. t 400000.9 t Page 4 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Graph: On the graphing calculator, graph 400000.9 and 700. Find the point of intersection. There will e 7,00 homes not connected to the sewer sstem after aout 8 ears. You Tr: A new car costs $,000. The value decreases % each ear. Write an eponential deca model for the car s value. Use the model to estimate the value of the car after ears. QOD: Descrie the end ehaviors of an eponential deca function. Sample CCSD Common Eam Practice Question(s): A compan commits to reducing its caron emissions 0% each ear from the preceding ear. If the compan emitted 0,000 tons of caron dioide in the ear prior to starting the program, which formula for cn represents the compan s emissions during ear n of the program? A. cn 00000.9 n n B. cn 00000. C. cn 0.90000 n D. cn 0000 0. n Sample SAT Question(s): Taken from College Board online practice prolems. n t 000.8 t The function aove can e used to model the population of a certain endangered species of animal. If n t gives the numer of the species living t decades after the ear 900, which of the following is true aout the population of the species from 900 to 90? (A) It increased aout,000. (B) It increased aout 0. (C) It decreased aout 80. (D) It decreased aout 0. (E) It decreased aout,000. Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eploration: Evaluate the following for larger and larger values of n. n Note: Students should get values closer and closer to.788, which is the value of e. Now use the e ke on the calculator, and it will automaticall give an approimation of e. Euler Numer: the irrational natural ase e.7888849 n Note: e is named after its discoverer, Leonhard Euler. Simplifing Epressions with Base e E: Simplif the epression Use the product of powers propert. 4e e. 4e e 4e 4e 7 e E: Simplif the epression e 4. Use the power of product and power of power properties. Use the quotient of powers propert. e 9e e e 4 8 9e 9e 8 9 9 Use the definition of negative eponents. 9 e Graphing Natural Base Functions E: Graph the functions e and e. State the domain and range. e e Eponential Growth Eponential Deca Domain: All real numers. Range: 0 Domain: All real numers. Range: 0 Page 6 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions E: Sketch the graph of the function. State its domain and range. Graph the function e. Because the coefficient of e is, we will reflect over the -ais. Then the graph will e shifted left and up. e 0 Domain: All real numers Range: -0-0 - -0 Continuousl Compounded Interest nt r Recall: A P is the formula for interest compounded n times per ear. If interest is n compounded continuousl, then n. Therefore, the formula for continuousl compounded rt interest is A Pe. E: Chelli deposited $00 into an account that pas 7.% annual interest compounded continuousl. What is her alance after ears? Use A rt Pe with 00 P, r 0.07, and t. A 0.07 00 e $74.7 You Tr: The atmospheric pressure P (in pounds per square inch) of an oject d miles aove sea 0.d level can e modeled P 4.7e. How much pressure per square inch would ou eperience at the summit of Mount Washington, 688 feet aove sea level? Graph the model and estimate our height aove sea level if ou eperience. l/in of pressure. QOD: e is an irrational numer. Define irrational and give other eamples of irrational numers. Page 7 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will evaluate and simplif epressions using properties of logarithms. Definition of a Logarithm: Let and e positive numers, with. log if and onl if Note: Evaluating a logarithm is the same as finding an eponent. log is read log ase of. Writing Logarithmic Equations in Eponential Form E: Rewrite the following in eponential form.. log 8 The ase is, the eponent is, and the power is 8. 8. log The ase is, the eponent is, and the power is.. log0 0.0 The ase is 0, the eponent is, and the power is 0.0 Note: The logarithmic and eponential forms of the equations are equivalent. 0 0.0 Evaluating Logarithms: To evaluate a logarithmic epression, rememer ou are finding the eponent the ase would need to e raised to in order to otain the argument of the logarithm. E: Evaluate the following epressions.. log4 64 Answer the question: What eponent would I raise 4 to in order to otain 64? log4 64 ecause 4 64. log/ 9 Answer the question: What eponent would I raise / to in order to otain 9? log/ 9 ecause 9. log9 Answer the question: What eponent would I raise 9 to in order to otain? log9 ecause 9 Page 8 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Special Values of Logarithms: (Let e a positive real numer other than.). log Answer the question: What eponent would I raise to in order to otain? Because an positive real numer raised to the 0 power is equal to, we can sa that log 0.. log Special Logarithms Answer the question: What eponent would I raise to in order to otain? Because an numer raised to the power of is equal to itself, we can sa that log. Common Logarithm: the logarithm with ase 0 log0 log Natural Logarithm: the logarithm with ase e log ln E: Evaluate log00. Answer the question: What eponent would I raise 0 to in order to otain 00? log00 ecause 0 00 E: Evaluate ln e. e Answer the question: What eponent would I raise e to in order to otain e? ln e Logarithms on the Calculator: The calculator can onl evaluate common and natural logarithms. E: Approimate the value of log. E: Approimate the value of ln 0.0. You Tr: Evaluate the following logarithmic epressions.. log. log0.00. log/ 6 4. log6 7 QOD: Eplain wh the logarithm of a negative numer is undefined. Page 9 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential and logarithmic functions are inverse functions. Recall: B definition, if f and g are inverse functions, then f g g f Therefore, log and log E: Evaluate the epressions... ln 4 e ln 4 e 4. log0 log0. log Rewrite the argument to match the ase. Finding Inverses E: Find the inverse of the function log. log log The inverse of a logarithmic function is an eponential function. E: Find the inverse of the function ln. Step One: Switch the and the. ln Step Two: Solve for. Write in eponential form. e e Graphing Logarithmic Functions Recall: Graph the eponential function e. The graph of ln (the inverse of graph of e over the line. e ) is the reflection of the Domain: 0 ; Range: All real numers Asmptote: 0 Note: The domain and range of a logarithmic function is the domain and range of its inverse (eponential function) switched! Page 0 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions E: Graph the logarithmic function. State the domain and range. log Plot convenient points: Let. log log 0 6 6 6. Let 4 log 4 log 6 6 6 Because the graph is shifted to the left, there is an asmptote at. 0 - Domain: ; Range: All real numers - -0 E: Sketch the graph of the logarithmic function. State the domain and range. log/ Plot convenient points: Let. log/ Let. log/ Let. log/ 4 The graph is shifted down, and there is an asmptote at 0. Domain: 0 ; Range: All real numers 0 - - -0 E: The slope s of a each is related to the average diameter d (in millimeters) of the sand particles on the each the equation s 0.9 0.8log d. Graph the model and estimate the average diameter of the sand peles for a each whose slope is 0.. The average diameter is aout 0.84 millimeters. You Tr: Sketch the graph of f log. QOD: For what values of does the graph of f intersect its inverse Page of McDougal Littell: 8. 8.7 f log?
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sample CCSD Common Eam Practice Question(s): What graph represents the function f log? Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojectives: 8. The student will simplif an epression involving real-numer eponents using laws of eponents. 8. The student will evaluate and simplif epressions using properties of logarithms. Review: Properties of Eponents (Allow students to come up with these on their own.) We will now etend these properties for use with logarithms. Let a and e real numers, and let m and n e integers. Product of Powers Propert Quotient of Powers Propert m n m n a a a m m a mn a a or, a 0 n n nm a a a m Power of a Power Propert n a a mn Because of the relationship etween logarithms and eponents, the properties of logarithms are similar. Properties of Logarithms: Let, r, and v e positive numers with. Product Propert log uv log u log v u Quotient Propert log logu logv v n Power Propert log u nlog u Using Properties of Logarithms to Approimate the Value of a Logarithmic Epression E: Use the approimations log.46 and log7.77 to approimate the epressions. log Rewrite as a product: log 7. Use the product propert of logarithms. log 7 log log 7 Sustitute the values of the logarithms. log log7.46.77. 7 7. log Use the propert of logarithms. log log7 log Sustitute the values of the logarithms. log7 log.77.46 0. Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions. log Rewrite as a power: log Use the power propert of logarithms. log log.46.9 Sustitute the values of the logarithms. Rewriting Logarithmic Epressions E: Epand the epression 4 log z. Assume all variales are positive. Quotient Propert: Product Propert: 4 4 log log log z log log log z 4 z Power Propert: 4log log logz E: Write an equivalent form of the epression ln ln ln z. Assume all variales are positive. Power Propert: Quotient Propert: ln ln ln ln ln ln z z ln ln ln z ln ln z z Evaluating Logarithms of Base Start with log u. Rewrite in eponential form: u Take the log of oth sides (An log will do!): log log u We will use either the common or natural logarithm ecause these are the logs that the calculator can evaluate. Use the power propert: log log u Solve for : log u log Page 4 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Change of Base Formula: log u log log c c u E: Use the change of ase formula to approimate the value of log. log Using common logarithms: log.6 log ln Now tr the same eample using natural logarithms: log.6 ln Note: We approimated the value of this logarithm previousl in the notes. Compare! Graphing Logarithmic Functions: We can now use the change of ase formula to graph logarithmic functions of an ase on the graphing calculator. f log. E: Graph the function / Rewrite the function using the change of ase formula. (We will use the natural logarithm.) ln f log/ ln Note: We graphed this function hand earlier. Compare! You Tr: Epand the epression log 4 log log. Assume all variales are positive. QOD: Eplain how ou can rewrite the epression log log without using the quotient propert of logarithms. (Hint: Use the power and product properties.) Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sample CCSD Common Eam Practice Question(s): Epand the epression logn. A. loglogn B. log log n C. loglog n D. loglog n Page 6 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8.4 The student will solve eponential and logarithmic equations including ase e. Solving Eponential Equations Method : Rewrite oth sides of the equation so that the have the same ase. Note: If, then. E: Solve the equation 8 4. Rewrite oth sides with a ase of. (Note: Use the power of a power propert. Equate the eponents and solve for. 8 and ) 4 Check the solution sustituting into the original equation. Method : Taking a logarithm of oth sides. E: Solve the equation 8. Take the log ase of oth sides. log log8 Simplif using inverses. log 8 8 4 4 64 True Use the change of ase formula to evaluate. log8.89 log Alternate Method: Take the natural log of oth sides. ln ln8 Use the power propert of logs. ln ln8 Solve for. ln8.89 ln Check our answer using the calculator. Page 7 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions 4 E: Solve the equation 8. Isolate the eponential term. Take the common logarithm of oth sides. 4 7 log 4 log7 Use the power propert of logs. 4log log7 Solve for. log 7 4 log log 7 4 log log7 4 0. log Check the solution. Because of the compleit of the solution, a good wa to check would e to graph oth sides of the original equation on the graphing calculator and find the point of intersection. Solving a Logarithmic Equation Method : If the logarithms on oth sides have the same ase, use the fact that log onl if. E: Solve the equation log log 8 7. 4 4 log if and Both sides have a logarithm with ase 4, so we can equate the arguments. 8 7 Solve for. Check the solution in the original equation. 4 4 4 4 7 4 log log 8 7 log log Method : Rewriting the equation in eponential form. log 4. E: Solve the equation Rewrite in eponential form. 4 Page 8 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Solve for. 4 9 Check the solution in the original equation. Simplifing Before Solving in a Logarithmic Equation log 9 4 log True E: Solve the equation log log. 6 6 Rewrite the left side of the equation using the product propert of logs. 6 log Write the equation in eponential form. Solve for. 6 6 60 9 4 0 9,4 Check the solutions in the original equation. 9 : log 9 log 9 Not possile to take the log of a negative numer! 6 6 4 : log 4 log 4 log 9 log 4 log 6 True 6 6 6 6 6 Solution: 4 (Note: 9 is an etraneous solution) You Tr: Solve the equations.. 4 8e 0.0t log log 7. QOD: Eplain wh logarithmic equations can have etraneous solutions. Page 9 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sample CCSD Common Eam Practice Question(s):. Which equation is equivalent to A. log4 8 log0 4 B. 8 log 0 C. log8 4 D. log8 4 4 8?. What is the solution of the equation A. k B. k C. k 4 D. k n. Solve the equation e 0 for n. A. 0 n ln e B. nln 0 e C. n ln0 ln0 D. n e log k? 4. What is the value of if 9? A. 4 B. C. D. Page 0 of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will develop mathematical models using eponential or logarithmic equations to solve real world prolems. Writing an Eponential Function: two points determine a unique eponential function a E: Write an eponential function a whose graph passes through, 7 and, 6. Sustitute each ordered pair in for and in the equation a : 7 a 6 a To eliminate a, divide the two equations. Put the highest power of on top: 6 a 7 a 9 Solve for : (Note: In an eponential function, cannot e negative.) 7a a Sustitute this value of into one of the original equations and solve for a: 7 Eponential Function: 7 Writing a Power Function: two points determine a unique power function a E: Write a power function a whose graph passes through,8 and 9,. Sustitute each ordered pair in for and in the equation Solve one of the equations for a: Sustitute the epression into the other equation for a: Use the power of a quotient propert to simplif: a : a 8 a 9 8 8 a a 8 9 9 8 8 log. Solve for using logarithms:. log. 0.69 log 8 8 Sustitute this value in for to solve for a: a. 0.69 Power Function:. 0.69 Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Using Eponential and Power Models E: Find an eponential model to fit the data. Use the model to estimate when is. 0 4 6 7 8 9 4.7..9.4.9.4 0.9 0.4 0.0 9.6 Enter the -values into L and the -values into L (Go to STAT Edit) On the Home screen, go to STAT CALC and choose option 0, EpReg. Then tpe Y (found in the VARS menu), and press Enter. This will calculate the eponential regression model and store it in Y. Graph the scatter plot along with the eponential regression equation to see if the model fits the data. Use ZoomStat. Eponential Model: 4.97 0.96 When is, 4.97 0.96 7.8 E: The ordered pairs tr, descrie the circular area r (square feet) that oil from a leaking oil tanker covers t minutes after it egins leaking. Find a power model for the data. Use the model to estimate the area that will e covered the leaking oil after hours. t 0 0 60 r 8.6 706. 86 68. 04 766 468. 076 Enter the t-values into L and the r-values into L (Go to STAT Edit) On the Home screen, go to STAT CALC and choose option A, PwrReg. Then tpe Y (found in the VARS menu), and press Enter. This will calculate the power regression model and store it in Y. Graph the scatter plot along with the eponential regression equation to see if the model fits the data. Use ZoomStat. Power Model: 8.8 After hours, t is 0 minutes, 8.8 0 4069. ft Page of McDougal Littell: 8. 8.7
Algera II Notes Unit Eight: Eponential and Logarithmic Functions You Tr:. Write an eponential function of the form a that passes through the points,6 and,8. Check our answer using the eponential regression function on the graphing calculator.. Write a power function of the form a that passes through the points 0., and 0,0. Check our answer using the power regression function on the graphing calculator. QOD: Which values are constant in an eponential function? In a power function? Page of McDougal Littell: 8. 8.7