exponential and logarithmic functions
TRANSCRIPT
EXPONENTIAL AND LOGARITHMIC
FUNCTIONSPRECALCULUS
CHAPTER 3
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β’ This Slideshow was developed to accompany the textbook
β’ Precalculus
β’ By Richard Wright
β’ https://www.andrews.edu/~rwright/Precalculus-RLW/Text/TOC.html
β’ Some examples and diagrams are taken from the textbook.
Slides created by Richard Wright, Andrews Academy [email protected]
Slides created by Richard Wright, Andrews Academy [email protected]
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3-01 EXPONENTIAL FUNCTIONSIn this section, you will:
β’ Evaluate exponential functions with base b.
β’ Graph exponential functions with base b.
β’ Evaluate and graph exponential functions with base e.
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3-01 EXPONENTIAL FUNCTIONS
β’ Exponential function
β’ π¦ = π β ππ₯
β’ a is initial amount (y-int)
β’ b is base
β’ x is exponent
β’ If b > 1
β’ Exponential Growth
β’ If 0 < b < 1
β’ Exponential Decay
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3-01 EXPONENTIAL FUNCTIONS
β’ Domain: All real numbers
β’ Range: (0, β)
β’ Horizontal Asymptote:
β’ y = 0
β’ y-intercept: (0, 1)
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3-01 EXPONENTIAL FUNCTIONS
β’ Transformations
β’ π¦ = π β ππ₯ββ + π
β’ a vertical stretch
β’ If a is negative, then reflected over x-axis
β’ h moves right
β’ k moves up
β’ Domain: All Real
β’ Range:
β’ (k, β) if a > 0
β’ (-β, k) if a < 0
β’ HA: y = k
β’ y-int: (0, a + k) if h = 0
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3-01 EXPONENTIAL FUNCTIONS
β’ Graph by making a table
β’ Graph π¦ = 4βπ₯ + 3
β’ Decay
β’ HA: y = 3
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3-01 EXPONENTIAL FUNCTIONS
β’ Exponential functions are one-to-one
β’ Each x gives a unique y
β’ Solve 16 = 2π₯+2
16 = 2π₯+2
24 = 2π₯+2
Exponents must be equal4 = π₯ + 22 = π₯
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3-01 EXPONENTIAL FUNCTIONS
β’ Solve 1
3
π₯= 81
1
3
π₯
= 34
1
3
π₯
=1
3
β4
Exponents must be equalπ₯ = β4
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3-01 EXPONENTIAL FUNCTIONS
β’ Natural Base
β’ e β 2.718281828β¦
β’ π π₯ = ππ₯
Slope of any tangent line to e^x is e^x
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3-01 EXPONENTIAL FUNCTIONS
β’ Compound Interest
β’ π΄ = π 1 +π
π
ππ‘
β’ A = current amount
β’ P = principle (initial amount)
β’ r = yearly interest rate (APR)
β’ n = number of compoundings per year
β’ t = years
β’ Compounded Continuously
β’ π΄ = ππππ‘
β’ π = 1 +1
π
π
β’ When π β β
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3-02 LOGARITHMIC FUNCTIONSIn this section, you will:
β’ Evaluate logarithmic functions with base b.
β’ Evaluate logarithmic functions with base e.
β’ Use logarithmic functions to solve real world problems.
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3-02 LOGARITHMIC FUNCTIONS
β’ π π₯ = logπ π₯
β’ βlog base b of xβ
β’ Logarithms are inverses of exponential functions
β’ π¦ = logπ π₯ β π₯ = ππ¦
β’ Logarithms are exponents!
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3-02 LOGARITHMIC FUNCTIONS
β’ Evaluate
β’ Think βWhat exponent of the base gives the big number?β
β’ log5 125
β’ log21
64
Think 5π₯ = 12553 = 125So log5 125 = 3
Think 2π₯ =1
64
2β6 =1
64
So log21
64= β6
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3-02 LOGARITHMIC FUNCTIONS
β’ Calculator
β’ LOG β log10 β log
β’ LN β logπ β ln
β’ Use your calculator to evaluate log 300
2.477
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3-02 LOGARITHMIC FUNCTIONS
β’ Properties of Logarithms
β’ logπ 1 = 0
β’ logπ π = 1
β’ logπ ππ₯ = π₯
β’ If logπ π₯ = logπ π¦, then π₯ = π¦
β’ Simplify
β’ log5 1
β’ logπ π
β’ 8log8 30
log5 1 = 0
logπ π = ln π = 1
8log8 30 = π₯Rewrite as an exponential
log8 π₯ = log8 30π₯ = 30
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3-02 LOGARITHMIC FUNCTIONS
β’ Solve
β’ log3 π₯2 + 4 = log3 29
Since logs are the sameπ₯2 + 4 = 29π₯2 = 25π₯ = Β±5
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3-03 PROPERTIES OF LOGARITHMSIn this section, you will:
β’ Use properties of logarithms to expand logarithmic expressions.
β’ Use properties of logarithms to condense logarithmic expressions.
β’ Use the change-of-base formula to evaluate logarithms.
β’ Graph logarithmic functions.
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3-03 PROPERTIES OF LOGARITHMS
β’ Properties of Logarithms
β’ Product Property: logπ π’π£ = logπ π’ + logπ π£
β’ Quotient Property: logππ’
π£= logπ π’ β logπ π£
β’ Power Property: logπ π’π = π logπ π’
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3-03 PROPERTIES OF LOGARITHMS
β’ Write each log in terms of ln 2 and ln 5.
β’ ln 10
β’ ln5
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ln 10ln 2 β 5ln 2 + ln 5
ln5
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ln5
25
ln 5 β ln 25
ln 5 β 5 ln 2
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3-03 PROPERTIES OF LOGARITHMS
β’ Expand
β’ log 3π₯2π¦β’ ln
4π₯+1
8
log 3 + log π₯2 + log π¦log 3 + 2 log π₯ + logπ¦
ln 4π₯ + 112 β ln 8
1
2ln(4π₯ + 1) β ln 8
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3-03 PROPERTIES OF LOGARITHMS
β’ Condense
β’1
3log π₯ + 5 log π₯ β 3
β’ 4 ln π₯ β 4 β 2 ln π₯
log π₯13 + log π₯ β 3 5
log π₯13 π₯ β 3 5
ln π₯ β 4 4 β ln π₯2
lnπ₯ β 4 4
π₯2
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3-03 PROPERTIES OF LOGARITHMS
β’ Condense
β’1
5log3 π₯ + log3 π₯ β 2
1
5log3 π₯ + log3 π₯ β 2
1
5log3 π₯ π₯ β 2
log3 π₯ π₯ β 215
log35π₯ π₯ β 2
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3-03 PROPERTIES OF LOGARITHMS
β’ Change-of-Base Formula
β’ logπ π =logπ π
logπ π
β’ Evaluate
β’ log3 17
log 17
log 3= 2.579
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3-03 PROPERTIES OF LOGARITHMS
β’ Because logs are inverses of exponentials, the x and y is switched and the graph is flipped over the line y = x.
β’ π¦ = logπ(π₯ β β)
β’ Domain: π₯ > β
β’ Range: all real
β’ VA: π₯ = β
β’ x-int: (h + 1, 0)
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3-03 PROPERTIES OF LOGARITHMS
β’ To graph a logarithm
β’ Find and graph the vertical asymptote
β’ Make a table
β’ Use change-of-base formula
β’ logπ π₯ =log π₯
log π
β’ Or use the logBASE function on some TI graphing calcs
β’ MATH β logBASE
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3-03 PROPERTIES OF LOGARITHMS
β’ Graph π¦ = log2(π₯ + 1)
Change-of-base gives π¦ = log2 π₯ + 1 β π¦ =log π₯+1
log 2
x | y-1 | Error0 | 01 | 12 | 1.583 | 24 | 2.325 | 2.586 | 2.817 | 3
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONSIn this section, you will:
β’ Use one-to-one property to solve exponential equations.
β’ Use one-to-one property to solve logarithmic equations.
β’ Solve general exponential equations.
β’ Solve general logarithmic functions.
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
β’ Solve Exponential Equations
β’ Shortcut Method
β’ 1-to-1 method (rewrite with the same base)
β’1
5
π₯= 125
1
5
π₯
= 125
1
5
π₯
=1
5
β3
π₯ = β3
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
β’ General Method
β’ Take log of both sides
β’ 5 β 3ππ₯ = 2
β’ 6 2π‘+5 + 4 = 11
5 β 3ππ₯ = 2β3ππ₯ = β3ππ₯ = 1
ln ππ₯ = ln 1π₯ = 0
6 2π‘+5 + 4 = 116 2π‘+5 = 7
2π‘+5 =7
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log2 2π‘+5 = log2
7
6
π‘ + 5 = log27
6
π‘ = β5 + log27
6β β4.778
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
β’ π2π₯ β 7ππ₯ + 12 = 0
ππ₯ β 3 ππ₯ β 4 = 0ππ₯ β 3 = 0 ππ₯ β 4 = 0
ππ₯ = 3 ππ₯ = 4ln ππ₯ = ln 3 ln ππ₯ = ln 4
π₯ = ln 3 β 1.099 π₯ = ln 4 β 1.386
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
β’ Logarithmic Equations
β’ Shortcut Method
β’ 1-to-1 Property
β’ ln π₯ β ln 3 = 0
ln π₯ β ln 3 = 0ln π₯ = ln 3π₯ = 3
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
β’ General Method
β’ Exponentiate both sides
β’ 6 + 3 ln π₯ = 4
β’ log4 π₯ + log4(π₯ β 9) = 1
6 + 3 ln π₯ = 43 ln π₯ = β2
ln π₯ = β2
3
πln π₯ = πβ23
π₯ = πβ23 β 0.513
log4 π₯ + log4(π₯ β 9) = 1log4 π₯ π₯ β 9 = 1
4log4 π₯ π₯β9 = 41
π₯ π₯ β 9 = 4π₯2 β 9π₯ β 4 = 0
π₯ =9 Β± 92 β 4 1 β4
2 1
π₯ =9 Β± 97
2β 9.424,β0.424
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3-04 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
β’ Graphical method
β’ If the other methods donβt apply
β’ Make = 0
β’ Find the x-int
β’ Solve log2 π₯ = ln 2π₯
log2 π₯ β ln 2π₯ = 0Graph and find x-int
π₯ = 4.786
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3-05 EXPONENTIAL AND LOGARITHMIC MODELSIn this section, you will:
β’ Use exponential growth and decay models.
β’ Use the Gaussian model.
β’ Use the logistic growth model.
β’ Use logarithmic models.
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
Exponential Growth π¦ = ππππ₯ Exponential Decay π¦ = ππβππ₯
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
β’ Suppose a population growing according to the model π = 800π0.03π‘
where t is in years.
β’ What is the initial size?
β’ How long to double?
Let π‘ = 0.
π = 800π0.03 0 = 800
1600 = 800π0.03π‘
2 = π0.03π‘
ln 2 = ln π0.03π‘
ln 2 = 0.03π‘π‘ = 23.10 π¦ππ
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
β’ Radioactive decay
β’ π¦ = ππβππ₯
β’ π΄ = π΄0πππ‘
β’ Half-life
β’ Time it takes for Β½ of the material to decay
Very complicated, but we will use a simple modelThe dating method depends on the initial conditions. These are not really known for prehistorical situations.
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
β’ C14 has a half-life of 5700 years. If a sample starts with 3 g of C14, how much will remain after 100 years?
Find decay constant
π΄ = π΄0πππ‘
1.5 = 3ππ 5700
1
2= ππ 5700
ln1
2= ln ππ 5700
ln1
2= π 5700
π β β1.216 Γ 10β4
Find model
π΄ = 3π β1.216Γ10β4 π‘
Plug in 100 years
π΄ = 3π β1.216Γ10β4 (100) β 2.97 π
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
β’ Gaussian Model βThe Curveβ
β’ Normal Distribution
β’ π¦ = ππβπ₯βπ 2
π
Average
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
β’ Logistic Growth Model
β’ Used for population
β’ π¦ =π
1+ππβππ₯
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3-05 EXPONENTIAL AND LOGARITHMIC MODELS
β’ Logarithmic Models
β’ π¦ = π + π ln π₯
β’ π¦ = π + π log π₯
β’ Richter Scale
β’ Earthquake magnitude
β’ Decibels
β’ Loudness of sound
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