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C HAPTER 3 – E XPONENTIAL, L OGISTIC, AND LOGARITHMIC FUNCTIONS Overview: interrelationships between exponential, logistic, and logarithmic functions. Polynomial, rational, and power functions with rational exponents are algebraic functions. In this chapter, we will explore transcendental functions.

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T RANSCENDENTAL F UNCTIONS : Exponential functions- model growth and decay over time Ex: unrestricted population growth and the decay of radioactive substances Logistic functions- model restricted population growth, certain chemical reactions, and the spread of rumors and diseases. Logarithmic functions- are the basis of the Richter scale of earthquake intensity, the pH acidity scale, and the decibel measurement of sound.

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3.1 E XPONENTIAL AND LOGISTIC F UNCTIONS Section 3.1 #1-10e, 11, 12

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T HE FUNCTIONS F ( X )= X 2 AND G ( X )=2 X EACH INVOLVE A BASE RAISED TO A POWER, BUT THEIR ROLES ARE REVERSED : F(x)=x 2, the base is a variable x, the exponent is constant 2; F(x) is a ___________ and ___________________. G(x)=2 x, the base is constant 2, the exponent is a variable x; G(x) is an exponential function. DEFINITION: Let a and b be real number constants. An exponential function in x is a function that can be written in the form F(x)=a*b x, Where a is nonzero, b is positive, and b≠1. The constant a is the initial value of f (the value at x=0), and b is the base.

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I DENTIFYING EXPONENTIAL FUNCTIONS : F(x)=4 x G(x)=7x -5 H(x)=-3*3.5 x K(x)=6*2 -x Q(x)=3*7

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F INDING AN E XPONENTIAL F UNCTION FROM ITS TABLE OF VALUES XG(x)H(x) -24/9128 4/332 048 1122 2361/2

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E XPONENTIAL GROWTH AND DECAY For any exponential function f(x)=a*b x and any real number x, F(x+1)=b*f(x) If a>0 and b>1, the function f is increasing and is an exponential growth function. The base b is its growth factor. If a>0 and b<1, f is decreasing and is an exponential decay function. The base b is its decay factor.

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T RANSFORMING EXPONENTIAL FUNCTIONS F ( X )=2 X G(x)=2 x-3 H(x)=2 -x K(x)=3*2 x

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T HE N ATURAL BASE The functions f(x)=e x is one of the basic functions, exponential growth function. EXPONENTIAL FUNCTIONS AND THE BASE e Any exponential function f(x)=a*b x can be written as F(x)=a*e kx For an appropriately chosen real number constant k. If a>0 and k>0, f(x)=a*e kx is an exponential growth function. If a>0 and k<0, f(x)=a*e kx is an __________________ function.

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T RANSFORMING E XPONENTIAL F UNCTIONS F(x)=e x G(x)=e 2x H(x)=e -x K(x)=3e x

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DO NOW: If a>0, how can you tell whether y=a*b x represents an increasing or decreasing function?

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L OGISTIC F UNCTIONS AND T HEIR G RAPHS Let a, b, c, and k be positive constants, with b<1. A logistic growth function in x is a function that can be written in the form F(x)= c/(1+a*b x ) or f(x)= c/(1+a*e -kx ) Where the constant c is the “limit” to growth If b>1 or k<0, these formulas are logistic __________ functions.

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BASIC FUNCTION: T HE L OGISTIC F UNCTION F(x)=1/(1+e -x ) Domain: Range: Continuous? Bounded? Extrema? Horizontal Asymptotes: Vertical Asymptotes: Limit to the left: Limit to the right:

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G RAPHING L OGISTIC G ROWTH F UNCTIONS (a) f(x)= 8/(1+3*0.7 x ) (b) g(x)= 20/(1+2*e -3x )

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M ODELING H OBOKEN ’ S P OPULATION YearPopulation 1990 33,397 2000 38,577 Assuming the growth is exponential, when will the population of Hoboken surpass 1 million people? (HINT: Let P(t) be the population of Hoboken t years after 1990. Because P is exponential, P(t)=P 0 *b t, where P 0 is the initial (1990) population of 33,397)

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D ALLAS ’ S P OPULATION Based on recent data, a logistic model for the population of Dallas, t years after 1900, is: P(t)= 1,301,642/(1+21.602e -0.05054t ) According to this model, when was the population 1 million?

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DO NOW: If you were to get paid a quarter on the first day of the month, fifty cents on second day, one dollar on the third day, and this pattern continues throughout the month how much would you get paid on day 23 of the month? On day 30?

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3.2 – E XPONENTIAL AND L OGISTIC M ODELING HW: Pg 296 #7-20e For extra credit: #28

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I F A CULTURE OF 100 BACTERIA IS PUT INTO A PETRI DISH AND THE CULTURE DOUBLES EVERY HOUR, HOW LONG WILL IT TAKE TO REACH 400,000? I S THERE A LIMIT TO GROWTH ? Time (in hours)Bacteria 0 1 2 3 4

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E XPONENTIAL P OPULATION M ODEL If a population P is changing at a constant percentage rate r each year, then P(t)=P 0 (1+r) t Where P 0 is the initial population, r is expressed as a decimal, and t is time in years. If r>0, then P(t) is an exponential growth function, and its growth factor is the base of the exponential function, 1+r.

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F INDING G ROWTH AND D ECAY R ATES Tell whether the population model is an exponential growth function or exponential decay function, and find the constant percentage rate of growth or decay. (a) San Jose: P(t)=782,248*1.0136 t (b) Detroit: P(t)=1,203,368*0.9858 t

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D ETERMINE THE EXPONENTIAL F UNCTION THAT SATISFIES THE GIVEN CONDITIONS : Initial value=5, increasing at a rate of 17% per year Initial value=16, decreasing at a rate of 50% per month Initial population= 28900, decreasing at a rate of 2.6% per year Initial mass = 0.6 g, doubling every 3 days Initial mass= 592g, halving once every 6 years

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M ODELING T WO S TATES ’ P OPULATIONS U SING L OGISTIC R EGRESSION The populations (in millions) of Florida and Pennsylvania is represented with models: F(t)=28.021/(1+9018.63 -0.047015t ) P(t)=12.579/(1+29.0003e -0.034315t ) What are the limits?

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M ODELING A R UMOR Forks High School has 1200 students. Eric, Jessica, Mike, and Angela start a rumor that Edward and Bella are dating, which spreads logistically so that S(t)=1200/(1+39*e -0.9t ) models the number of students who have heard the rumor by the end of t days, where t=0 is the day the rumor begins to spread. (a) How many students have the rumor by the end of Day 0? (b) How long does it take for 1000 students to hear the rumor?

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N AME THE T YPE OF F UNCTION, T HEN FIND DETERMINE THE FUNCTION WITH THE GIVEN VALUES :

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3.3 – L OGARITHMIC F UNCTIONS AND T HEIR G RAPHS HW: Pg. 308 #1-18e

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C HANGING B ETWEEN L OGARITHMIC AND E XPONENTIAL F ORM If x>0 and 0 < b≠0, then Y=log b (x) IFF b y =x (a) log 2 8=3 because 2 3 =8 (b) log 3 √3=1/2 because 3 1/2 =√3 (c) log 5 1/25 = -2 because 3 -2 =1/5 -2 =1/25 (d) log 4 1=___ because 4 __ = 1 (e) log 7 7=___ because 7 __ = 7

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B ASIC P ROPERTIES OF L OGARITHMS For 0 0, and any real number y. log b 1=___ because b log b b=___ because b log b b y =___ because b = b y B log b x =___ because log b x=

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E VALUATING L OGARITHMIC AND E XPONENTIAL E XPRESSIONS LOG B B Y = Y (a) log 2 8= (b) log 3 √3= (c) 6 log 6 11 =

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xF(x)=2 x -2¼ ½ 01 12 24 xF -1 (x)=log 2 x ¼-2 ½ 10 21 42

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C OMMON L OGARITHMS -B ASE 10 Logarithms with base 10 are called common logarithms. The common logarithmic function log 10 x=log x, which is the inverse of the exponential function f(x)=10 x So y=logx IFF 10 y =x

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B ASIC P ROPERTIES OF C OMMON L OGARITHMS Let x and y be real numbers with x>0. Log 1=0 because 10 0 =1 Log 10=1 because 10 1 =10 Log 10 y =y because 10 y =10 y 10 Logx =x because log x = log x

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E VALUATE L OGARITHMIC AND E XPONENTIAL E XPRESSIONS -B ASE 10 Log100= Log 5 √10= Log1/1000= 10 log6 = HW: Pg.308 # 33-52e

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E VALUATING C OMMON L OGARITHMS WITH A C ALCULATOR (a) log 36.5 = (b) log.46 = Solving Simple Logarithmic Equations Log x = 4 Log 2 x= 5

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N ATURAL L OGARITHMS – B ASE E We often use the special abbreviation “ln” to denote a natural logarithm. Log e x=lnx Y = ln x IFF _______

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B ASIC P ROPERTIES OF N ATURAL L OGARITHMS Let x and y be real numbers with x>0 Ln 1 = 0 because ______ Ln e = __ because _______ Ln e y = ___ because ________ e lnx = x because ln x = ln x

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E VALUATE L OGARITHMS : Ln√e = Ln e 5 = e ln4 = Ln 23.5 = Ln 0.5 =

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T HE N ATURAL L OGARITHMIC F UNCTION F(x)=ln x Domain: Range: Continuous? Inc? Dec? Symmetry? Bounded? Extrema? H.A? V.A? End Behavior:

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T RANSFORMING L OGARITHMIC G RAPHS Y = LN X OR Y = LOG X G(x)= ln (x+4) G(x)= ln (5-x) H(x)= 6 log x H(x)= 8 + log x

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3.4 – P ROPERTIES OF L OGARITHMIC F UNCTIONS Pg. 317 #18-36e

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P ROPERTIES OF L OGARITHMS Let b, R, and S be positive real numbers with b≠1, any real number. Product rule: log b (RS) = log b R + log b S Quotient rule: log b (R/S) = log b R – log b S Power rule: log b R c = c log b R

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E XPANDING THE L OGARITHM OF A P RODUCT Log (8xy 4 ) = Ln √(x 2 +5)/x

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C ONDENSING A L OGARITHMIC E XPRESSION Lnx 4 – 3lnxy 6lnx – 4ln3x

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C HANGE OF B ASE Log 4 7 Let y = log 4 7 4 y =7

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C HANGE -O F -B ASE F ORMULA FOR L OGARITHMS For positive real numbers a, b, and x with a≠1 and b≠1. Log b x = log a x/log a b Log b x = logx/logb OR Log b x = lnx/lnb

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E VALUATE L OGARITHMS BY CHANGING THE B ASE Log 4 17= Log 2 10= Log 1/2 2=

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G RAPHS OF L OGARITHMIC F UNCTIONS WITH B ASE B We can rewrite any logarithmic function g(x)=Log b x as: G(x)=lnx/lnb=(1/lnb)lnx So every logarithmic function is a constant multiple of the natural log function f(x)=lnx If the base is b>1, the graph of g(x)=Log b x is vertical stretch or shrink of the graph of f(x)=lnx by the factor 1/lnb. If 0**
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D ESCRIBE HOW TO TRANSFORM THE GRAPH F ( X )=L NX INTO : (a) G(x)=Log 5 x (b) H(x)=Log 1/4 x

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3.5 –E QUATION S OLVING AND M ODELING HW: Pg. 331 #1-30e

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S OLVING E XPONENTIAL E QUATIONS One-To-One Properties For any exponential function f(x)=b x, If b u =b v, then u=v. For any logarithmic function f(x)=log b x If log b u=log b v, then u=v

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S OLVING AN E XPONENTIAL E QUATION A LGEBRAICALLY Solve 20(1/2) x/3 = 5

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S OLVE AN E XPONENTIAL E QUATION : (e x -e -x )/2=5

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S OLVE A L OGARITHMIC E QUATION : Logx 2 =2

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S OLVE L N (3 X -2) + L N ( X -1)=2L NX

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3.6 – M ATHEMATICS O F F INANCE

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S UPPOSE A PRINCIPAL OF P DOLLARS IS INVESTED IN AN ACCOUNT BEARING AN INTEREST RATE R EXPRESSED IN DECIMAL FORM AND CALCULATED AT THE END OF EACH YEAR. Time in YearsAmount in the Account OA o =P=principal 1A 1 =P+P*r=P(1+r) 2A 2 =A*(1+r)=P(1+r) 2 nA=A n =P(1+r) n

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S UPPOSE W EEHAWKEN T OWNSHIP INVESTS $500 AT 7% INTEREST COMPOUNDED ANNUALLY. F IND THE VALUE OF W EEHAWKEN ’ S INVESTMENT 10 YEARS LATER. Let P=_______, r=________, and n=________

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I NTEREST C OMPOUNDED K T IMES PER Y EAR Suppose a principal P is invested at an annual interest rate r compounded k times a year for t years. Then r/k is the interest rate per compounding period and kt is the number of compounding periods. The amount A in the account after t years is A=P(1+r/k) kt

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S UPPOSE J ENNIFER INVESTS $500 AT 9% ANNUAL INTEREST COMPOUNDED MONTHLY ( COMPOUNDED 12 TIMES A YEAR ). F IND THE VALUE OF HER INVESTMENT 5 YEARS LATER. Let P=______, r=_______, k=_____, and t=_____

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J ONATHON HAS $500 TO INVEST AT 9% ANNUAL INTEREST COMPOUNDED MONTHLY. H OW LONG WILL IT TAKE FOR HIS INVESTMENT TO GROW TO $3000?

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W ORK WITH A P ARTNER : Pg. 341 #1-8, 21-22

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Copyright © 2004 Pearson Education, Inc. Chapter 4 Exponential and Logarithmic Functions.

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