Name: Pre-calculus Guided Notes: Chapter 11 Epoetial ad Logarithmic Fuctios Sectio 2 Epoetial Fuctios Paret Fuctio: y = > 1 0 < < 1 Domai Rage y-itercept ehavior Horizotal asymptote Vertical asymptote Eample 1 Graph the epoetial fuctios y = 2, y = 2 + 3, ad y = 2-2 o the same set of aes. Compare ad cotrast the graphs. 1
Eample 2 1 Graph the epoetial fuctios y 3 Compare ad cotrast the graphs., 1 y 5, ad 3 1 y 3 o the same set of aes. Eample 3 A car depreciates or loses value at the rate of 20% per year. If the car origially cost $20,000, the depreciatio ca e modeled y the equatio y = 20,000(0.8) t, where y is the depreciatio ad t is the time i years. a. Fid the value of the car at the ed of 2 years.. Graph the depreciatio fuctio. 2
Whe a real-life quatity icreases or decreases y a fied percet each year (or other time period), the amout y of the quatity after t years ca e modeled y oe of the followig equatios: Epoetial Growth Model Epoetial Decay Model I these equatios, a = r = 1 + r = 1 r = Eample 4 The average growth rate of the populatio of a city is 7.5% per year. If the city s populatio is curretly 22,750 people, what is the epected populatio i 10 years? Compoud Iterest A P 1 r t P = r = = t = A = Eample 5 How much should Saria ivest ow i a moey market accout if she wishes to have $9000 i the accout at the ed of 10 years? The accout provides a APR of 6% compouded quarterly. 3
Sectio 3 The Numer e 10 1 10 2 10 3 10 4 10 5 10 6 1 1 As gets igger ad igger (approaches ), 1 1 approaches. This umer is called or ad is deoted. r You leared efore that the alace of a accout earig compoud iterest is give y A P 1. As the frequecy of compoudig approaches positive ifiity, the compoud iterest formula approimates the followig formula: Cotiuously Compouded Iterest A = Pe rt t A = r = P = t = Eample 1 Compare the alace after 30 years of a $15,000 ivestmet earig 12% iterest compouded cotiuously to the same ivestmet compouded quarterly. 4
Sectios 4, 5 ad 6 Logarithmic Fuctios (Icludig Commo ad Natural) Defiitio of Logarithm with Base Let ad y e positive umers with 1. The arithm of y with ase is deoted y y ad is defied as follows: if ad oly if Logarithmic Form Epoetial Form The epressio y is read as. Eample 1 Write each equatio i epoetial form. a. 1 27 3. 3 1 16 4 2 Eample 2 Write each equatio i arithmic form. a. 2 10 = 1024. 1 2 3 8 Eample 3 Evaluate. a.) 464.) 381 c.) 1/4256 d.) 100.001 e.) 642 f.) 366 5
Properties of Logarithms Let, m, ad e positive umers such that 1. Product Property Quotiet Property Power Property Equality Chage of Base Formula Special Logarithms Commo Logarithm m m m m m m If m =, the m = a a Natural Logarithm Eample 4 Solve each equatio. a. 1 1 6 15625. (2 + 5) = (5 4) 3 c. 3(4 + 5) 3(3 2) = 2 d. 11 + 11( + 1) = 116 6
Eample 5 Give that 5 = 0.6990, evaluate each arithm. a. 50,000. 0.005 Give that 2 = 0.3010, 3 = 0.4771, ad 7 = 0.8451, evaluate each arithm. a. 20,000. 14 c. 18 Eample 6 Evaluate. a. 91043. 315 Eample 7 Solve each equatio algeraically. a. 5 4 = 73. 2-1 = 5 2 c. 18 = e 3 d. 25e < 100 7
Graphig Logarithmic Fuctios You ca use the iverse relatioship etwee epoetial ad arithmic fuctios to graph arithmic fuctios. Eample 5 Graph each fuctio. a.) y = 3.) y = 2 ( + 3) + 1 y y 8
Sectio 7 Modelig Real-World Data with Epoetial ad Logarithmic Fuctios Eample 1 Be, a seior at BHS, has saved $2000 from his summer jo mowig laws. He would like to ivest the moey so that he will have doule the moey i si years whe he graduates from college. Be ivests the $2000 i a accout that pays 8% iterest compouded cotiuously. Will Be have douled his moey i 6 years? Eplai. If his ivestmet is t douled, what iterest rate would e ecessary i order for it to doule? Rememer all of that regressio stuff we used earlier this year, well you ca also apply it to epoetial ad arithmic data. Keep i mid the geeral shapes, i order to choose the est model. Eample 2 Idia is epected to have the largest populatio i the world y 2050. The tale elow gives the populatio of Idia i 100 millios for selected years durig the 1900s. Year Years Sice Populatio 1950 1950 3.58 1960 4.42 1970 5.55 1980 6.89 1990 8.51 a. Fid a equatio that models the populatio data show.. Use the equatio to predict the populatio of Idia i 2050. 9
Eample 3 The umer of acteria i a culture is oserved for several hours with the followig results recorded. Hours (t) 2.5 5.4 6.5 9.2 9.5 11.0 Numer of Bacteria (N) (thousads/cc) 10.06 14.59 20.70 27.91 31.50 40.06 Fid a equatio that models the data. Eample 4 A ice skater egis to coast with a iitial velocity of 4 meters per secod. The tale elow gives the times required for the skater to slow dow to various velocities. Fid a equatio that models the data. velocity (m/s) 3.5 3 2.5 2 1.5 1 0.5 time (s) 2.40 5.18 8.46 12.48 17.66 24.95 37.43 Eample 5 The tale elow gives some Cosumer Price Ide (CPI) values from 1955 to 2003. Year CPI 1955 26.8 1965 31.5 1975 53.8 1985 107.6 1995 152.4 2003 184.0 Source: www.ls.gov Liearize the data. That is, make a tale with - ad l y-values, where is the umer of years sice 1955 ad y is the CPI. The make a scatter plot of the liearized data. l y Fid a epoetial model for the origial data. Use the epoetial model to predict the CPI i 2010. 10