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Chapter 3 – Exponential, Logistic, and logarithmic functions

Chapter 3 – Exponential, Logistic, and logarithmic functions. Overview : interrelationships between exponential, logistic, and logarithmic functions. Polynomial, rational, and power functions with rational exponents are algebraic functions.

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Chapter 3 – Exponential, Logistic, and logarithmic functions

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  1. Chapter 3 – Exponential, Logistic, 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.

  2. Transcendental Functions: • 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.

  3. 3.1 Exponential and logistic Functions Section 3.1 #1-10e, 11, 12

  4. The functions f(x)=x2 and g(x)=2x each involve a base raised to a power, but their roles are reversed: • F(x)=x2, the base is a variable x, the exponent is constant 2; F(x) is a ___________ and ___________________. • G(x)=2x, 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 exponentialfunctionin x is a function that can be written in the form F(x)=a*bx, 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.

  5. Identifying exponential functions: • F(x)=4x • G(x)=7x-5 • H(x)=-3*3.5x • K(x)=6*2-x • Q(x)=3*7

  6. Finding an Exponential Function from its table of values

  7. Exponential growth and decay • For any exponential function f(x)=a*bx 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.

  8. Transforming exponential functionsf(x)=2x • G(x)=2x-3 • H(x)=2-x • K(x)=3*2x

  9. The Natural base • The functions f(x)=ex is one of the basic functions, exponential growth function. • EXPONENTIAL FUNCTIONS AND THE BASE e • Any exponential function f(x)=a*bx can be written as • F(x)=a*ekx For an appropriately chosen real number constant k. If a>0 and k>0, f(x)=a*ekx is an exponential growth function. If a>0 and k<0, f(x)=a*ekx is an __________________ function.

  10. Transforming Exponential Functions • F(x)=ex • G(x)=e2x • H(x)=e-x • K(x)=3ex

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

  12. Logistic Functions and Their Graphs • 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*bx) 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.

  13. F(x)=1/(1+e-x) Domain: Range: Continuous? Bounded? Extrema? Horizontal Asymptotes: Vertical Asymptotes: Limit to the left: Limit to the right: BASIC FUNCTION: The Logistic Function

  14. Graphing Logistic Growth Functions • (a) f(x)= 8/(1+3*0.7x) • (b) g(x)= 20/(1+2*e-3x)

  15. Modeling Hoboken’s Population 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)=P0*bt, where P0 is the initial (1990) population of 33,397)

  16. Dallas’s Population • 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?

  17. 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?

  18. 3.2 – Exponential and Logistic Modeling HW: Pg 296 #7-20e For extra credit: #28

  19. If 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? Is there a limit to growth?

  20. Exponential Population Model • If a population P is changing at a constant percentage rate r each year, then • P(t)=P0(1+r)t • Where P0 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.

  21. Finding Growth and Decay Rates • Tell whether the population model is an exponential growth function or exponential decay function, and find the constant percentage rate of growth or decay. • San Jose: P(t)=782,248*1.0136t • Detroit: P(t)=1,203,368*0.9858t

  22. Determine the exponential Function 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

  23. Modeling Two States’ Populations Using Logistic Regression • 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?

  24. Modeling A Rumor • 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?

  25. Name the Type of Function, Then find determine the function with the given values:

  26. 3.3 – Logarithmic Functions and Their Graphs HW: Pg. 308 #1-18e

  27. Changing Between Logarithmic and Exponential Form • If x>0 and 0 < b≠0, then Y=logb(x) IFF by=x (a) log28=3 because 23=8 (b) log3√3=1/2 because 31/2=√3 (c) log51/25 = -2 because 3-2=1/5-2=1/25 (d) log41=___ because 4__ = 1 (e) log77=___ because 7__ = 7

  28. Basic Properties of Logarithms For 0<b≠1, x>0, and any real number y. • logb1=___ because b • logbb=___ because b • logbby=___ because b = by • Blogbx=___ because logbx=

  29. Evaluating Logarithmic and Exponential Expressions logbby=y • log28= • log3√3= • 6 log611=

  30. Common Logarithms-Base 10 • Logarithms with base 10 are called common logarithms. • The common logarithmic function log10x=log x, which is the inverse of the exponential function f(x)=10x • So y=logx IFF 10y=x

  31. Basic Properties of Common Logarithms • Let x and y be real numbers with x>0. • Log 1=0 because 100=1 • Log 10=1 because 101=10 • Log 10y=y because 10y=10y • 10Logx=x because log x = log x

  32. Evaluate Logarithmic and Exponential Expressions-Base 10 • Log100= • Log5√10= • Log1/1000= • 10log6= • HW: Pg.308 # 33-52e

  33. Evaluating Common Logarithms with a CalculatorEvaluating Common Logarithms with a Calculator • (a) log 36.5 = • (b) log .46 = Solving Simple Logarithmic Equations • Log x = 4 • Log2x= 5

  34. Natural Logarithms – Base e • We often use the special abbreviation “ln” to denote a natural logarithm. • Logex=lnx • Y = ln x IFF _______

  35. Basic Properties of Natural Logarithms • Let x and y be real numbers with x>0 • Ln 1 = 0 because ______ • Ln e = __ because _______ • Lney = ___ because ________ • elnx = x because ln x = ln x

  36. Evaluate Logarithms: • Ln√e = • Ln e5 = • eln4 = • Ln 23.5 = • Ln 0.5 =

  37. F(x)=ln x Domain: Range: Continuous? Inc? Dec? Symmetry? Bounded? Extrema? H.A? V.A? End Behavior: The Natural Logarithmic Function

  38. Transforming Logarithmic Graphsy = 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

  39. 3.4 – Properties of Logarithmic Functions Pg. 317 #18-36e

  40. Properties of Logarithms • Let b, R, and S be positive real numbers with b≠1, any real number. • Product rule: logb(RS) = logbR + logbS • Quotient rule: logb(R/S) = logbR – logbS • Power rule: logbRc = c logbR

  41. Expanding the Logarithm of a Product • Log (8xy4) = • Ln √(x2+5)/x

  42. Condensing a Logarithmic Expression • Lnx4 – 3lnxy • 6lnx – 4ln3x

  43. Change of Base • Log47 • Let y = log47 • 4y=7

  44. Change-Of-Base Formula for Logarithms • For positive real numbers a, b, and x with a≠1 and b≠1. • Logbx = logax/logab • Logbx = logx/logb • OR • Logbx = lnx/lnb

  45. Evaluate Logarithms by changing the Base • Log417= • Log210= • Log1/22=

  46. Graphs of Logarithmic Functions with Base b • We can rewrite any logarithmic function g(x)=Logbx 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)=Logbx is vertical stretch or shrink of the graph of f(x)=lnx by the factor 1/lnb. • If 0<b<1, a reflection across the x-axis is required as well

  47. Describe how to transform the graph f(x)=Lnx into: • G(x)=Log5x • H(x)=Log1/4x

  48. 3.5 –Equation Solving and Modeling HW: Pg. 331 #1-30e

  49. Solving Exponential Equations • One-To-One Properties • For any exponential function f(x)=bx, • If bu =bv , then u=v. For any logarithmic function f(x)=logbx • If logbu=logbv, then u=v

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