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Chapt. 9Chapt. 9Exponential and Exponential and
Logarithmic FunctionsLogarithmic Functions
Mall Browsing Time vsMall Browsing Time vsAverage Amount SpentAverage Amount Spent
$0.00
$50.00
$100.00
$150.00
$200.00
$250.00
1 2 3 4 5
Time at Mall (hours)
Avera
ge A
mo
un
t S
pen
t
Exponential FunctionExponential Function Amount spent as a function of time spentAmount spent as a function of time spent
f(x) = 42.2(1.56)f(x) = 42.2(1.56)xx where x is in hours where x is in hours
(source: International Council of Shopping Centers Research, 2006)(source: International Council of Shopping Centers Research, 2006)
Exponential Function:Exponential Function:
f(x) = bf(x) = bxx or y = b or y = bxx,,
where b > 0 and b where b > 0 and b ≠ ≠ 1 and x is in 1 and x is in RR
Comparison of Linear, Quadratic, Comparison of Linear, Quadratic, and Exponential Functionsand Exponential Functions
xx f(x) = 1.56xf(x) = 1.56x f(x) = 1.56xf(x) = 1.56x22 f(x) = 1.56f(x) = 1.56xx
11 1.561.56 1.561.56 1.561.56
22 3.123.12 6.246.24 2.432.43
33 4.684.68 14.0414.04 3.803.80
44 6.246.24 24.9624.96 5.925.92
55 7.807.80 39.0039.00 9.249.24
66 9.369.36 56.1656.16 14.4114.41
77 10.9210.92 76.4476.44 22.4822.48
88 12.4812.48 99.8499.84 35.0735.07
99 14.0414.04 126.36126.36 54.7254.72
1010 15.6015.60 156.00156.00 85.3685.36
1111 17.1617.16 188.76188.76 133.16133.16
1212 18.7218.72 224.64224.64 207.73207.73
1313 20.2820.28 263.64263.64 324.06324.06
1414 21.8421.84 305.76305.76 505.53505.53
Comparison of Linear, Quadratic, Comparison of Linear, Quadratic, and Exponential Functionsand Exponential Functions
Examples of Exponential FunctionExamples of Exponential Function
f(x) = 2f(x) = 2xx
g(x) = 10g(x) = 10xx
h(x) = 5h(x) = 5x+1x+1
j(x) = (1/2)j(x) = (1/2)xx - 1- 1
NOT Exponential FunctionsNOT Exponential Functions
f(x) = xf(x) = x22
Base, not exponent, is variableBase, not exponent, is variable g(x) = 1g(x) = 1xx
Base is 1Base is 1 h(x) = (-3)h(x) = (-3)xx
Base is negativeBase is negative j(x) = xj(x) = xxx
Base is variableBase is variable
Evaluating Exponential FunctionEvaluating Exponential Function
Given: f(x) = 42.2(1.56)Given: f(x) = 42.2(1.56)xx
How much will an average mall shopper spend How much will an average mall shopper spend after 3 hours?after 3 hours?
f(3) = 42.2(1.56)f(3) = 42.2(1.56)33
≈ 42.2(3.796)≈ 42.2(3.796) ≈ 160≈ 160
Graphing Exponential Function:Graphing Exponential Function:f(x) = 3f(x) = 3x + 1x + 1
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
f(x) 3 x̂
3 (̂x+1)
x 3^x 3^(x+1)
-5 0.00 0.01
-4 0.01 0.04
-3 0.04 0.11
-2 0.11 0.33
-1 0.33 1.00
0 1.00 3.00
1 3.00 9.00
2 9.00 27.00
3 27.00 81.00
4 81.00 243.00
5 243.00 729.00
Excel
Graphing Exponential Function:Graphing Exponential Function:f(x) = 2f(x) = 2xx
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
-3 -2 -1 0 1 2 3 4 5
x
f(x)
x f(x)
-3 0.13
-2 0.25
-1 0.50
0 1.00
1 2.00
2 4.00
3 8.00
4 16.00
5 32.00
Graphing Exponential Function:Graphing Exponential Function:f(x) = (1/2)f(x) = (1/2)xx
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
f(x)
x f(x)
-5 32.00
-4 16.00
-3 8.00
-2 4.00
-1 2.00
0 1.00
1 0.50
2 0.25
3 0.13
4 0.06
5 0.03
Characteristics of f(x) = bCharacteristics of f(x) = bxx
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
f(x) 2 x̂
(0.5) x̂
x 2^x (0.5)^x
-5 0.03 32.00
-4 0.06 16.00
-3 0.13 8.00
-2 0.25 4.00
-1 0.50 2.00
0 1.00 1.00
1 2.00 0.50
2 4.00 0.25
3 8.00 0.13
4 16.00 0.06
Characteristics of f(x) = bCharacteristics of f(x) = bxx
Domain of f(x) Domain of f(x) = {- = {- ∞∞ , , ∞∞}}
Range of f(x)Range of f(x)= (0, = (0, ∞∞ ) )
bbx x passes through (0, 1)passes through (0, 1) For b>1, rises to rightFor b>1, rises to right
For 0<b<1, rises to leftFor 0<b<1, rises to left bbxx approaches, but does approaches, but does
not touch, x-axis, (x-axis not touch, x-axis, (x-axis called an called an assymptoteassymptote))
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
f(x) 2 x̂
(0.5) x̂
Application: Compound InterestApplication: Compound Interest
Suppose:Suppose: A: amount to be receivedA: amount to be received
P: principalP: principalr: annual interest (in decimal)r: annual interest (in decimal)n: number of compounding periods per yearn: number of compounding periods per yeart: yearst: years
r ntr ntA(t) = P 1 + ---A(t) = P 1 + --- n n
Application: Compound InterestApplication: Compound Interest What would be the yield for the following What would be the yield for the following
investment?investment? P = 8000P = 8000
r = 7%r = 7%n = 12n = 12t = 6 yearst = 6 years
r ntr ntA(t) = P 1 + ---A(t) = P 1 + --- n n
0.07 (12)(6)0.07 (12)(6)A = (8000) 1 + --------A = (8000) 1 + -------- 12 12 ≈≈ $12,160.84 $12,160.84
Excel
Application: Continuous Application: Continuous CompoundingCompounding
A(t) = PeA(t) = Pert rt where e = 2.71828where e = 2.71828……
What is the yield with the following What is the yield with the following conditions?conditions?
P = 8000P = 8000r = 6.85%r = 6.85%n = 12n = 12t = 6 yearst = 6 years
A =A = (8000)e (8000)e(0.0685)6(0.0685)6
= $12,066.60 = $12,066.60
Natural Base Natural Base ee
Recall: Recall: A = P(1 + (r/n))A = P(1 + (r/n))ntnt
Given Given A = $1 A = $1 r = 100% r = 100% t = 1 year t = 1 year
ThenThenA = (1 + (1/n))A = (1 + (1/n))nn
What is A, as n gets What is A, as n gets larger and larger?larger and larger?
n (1+1/n)^n
1 2.00
2 2.25
3 2.37
4 2.44
5 2.49
6 2.52
7 2.55
8 2.57
9 2.58
10 2.59
11 2.60
12 2.61
13 2.62
14 2.63
15 2.63
16 2.64
Natural Base Natural Base ee
(1 + (1/n))(1 + (1/n))nn 2.718281827.. = 2.718281827.. = ee e e = Natural base= Natural base
(Euler’s number) (Euler’s number) (Base of natural logarithms) (Base of natural logarithms)
Important mathematical constantsImportant mathematical constants 00 11 ii ππ ee
Natural Base Natural Base eef(x) = (1 + 1/x)^x
0.00
0.50
1.00
1.50
2.00
2.50
3.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
x
f(x)
x (1 + 1/x)^x
1 2.00
2 2.25
3 2.37
4 2.44
5 2.49
6 2.52
7 2.55
8 2.57
9 2.58
10 2.59
11 2.60
12 2.61
13 2.62
14 2.63
15 2.63
2.718
Your TurnYour Turn Sketch a graph Sketch a graph
(on the same (on the same coordinate coordinate system)system)
1.1. f(x) = 3f(x) = 3xx
2.2. f(x) = -3f(x) = -3xx
3.3. f(x) = 3f(x) = 3-x-x
4.4. f(x) = -3f(x) = -3-x-x
5.5. f(x) = 3f(x) = 3x+1x+1
6.6. f(x) = 3f(x) = 3x-1x-1
Exponential Functions
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
-5 -4 -3 -2 -1 -0 1 2 3 4 5
x
f(x)
3 x̂
-3 x̂
3 -̂x
-3 -̂x
3 (̂x-1)
3 (̂x+1)
9.2 Composite & Inverse Functions9.2 Composite & Inverse Functions
Given: (Discount Sale)Given: (Discount Sale) Discount 1: f(x) = x – 300Discount 1: f(x) = x – 300 Discount 2: g(x) = 0.85Discount 2: g(x) = 0.85xx
Composition of f and g:Composition of f and g: (f (f oo g)(x) = f(g(x))g)(x) = f(g(x))
Apply g(x) firstApply g(x) first Then, apply f(x) to the resultThen, apply f(x) to the result
Composite FunctionComposite Function Given: (Discount Sale)Given: (Discount Sale)
Discount 1: f(x) = x – 300Discount 1: f(x) = x – 300 Discount 2: g(x) = 0.85xDiscount 2: g(x) = 0.85x
For x = 1400For x = 1400 What is f(g(x))?What is f(g(x))? What is g(f(x))?What is g(f(x))?
f(g(1400)) f(g(1400)) = f(0.85 = f(0.85 · · 1400) = f(1190)1400) = f(1190) = 1190 – 300 = 890 = 1190 – 300 = 890
g(f(1400)) g(f(1400)) = g(1400 -300) = g(1400 -300) = g(1100)= g(1100)= 0.85 = 0.85 · · 1100 = 9351100 = 935
Composite FunctionsComposite Functions Given:Given:
f(x) = 3x – 4f(x) = 3x – 4 g(x) = xg(x) = x22 + 6 + 6
Composition (f Composition (f oo g)(x)g)(x) f(g(x)) = f(xf(g(x)) = f(x22 + 6) + 6)
= 3(x = 3(x22 + 6) – 4 + 6) – 4 = 3x = 3x22 + 18 – 4 + 18 – 4 = = 3x3x2 2 + 14+ 14
Composition (g Composition (g oo f f)(x))(x) g(f(x)) = g(3x – 4)g(f(x)) = g(3x – 4)
= (3x – 4) = (3x – 4)22 + 6 + 6 = 9x = 9x22 – 24x + 16 + 6 – 24x + 16 + 6 = = 9x9x22 – 24x + 24 – 24x + 24
Your TurnYour Turn Given:Given:
f(x) = 5x + 6f(x) = 5x + 6 g(x) = xg(x) = x22 – 1 – 1
Find: (f Find: (f oo g)(x) g)(x) f(g(x)) = f(xf(g(x)) = f(x22 – 1) – 1)
= 5(x = 5(x22 – 1) + 6 – 1) + 6 = 5x = 5x22 + 1 + 1
Find: (g Find: (g oo f)(x) f)(x) g(f(x)) = g(5x + 6)g(f(x)) = g(5x + 6)
= (5x + 6) = (5x + 6)22 – 1 – 1 = 25x = 25x22 + 60x + 36 – 1 + 60x + 36 – 1 = 24x = 24x22 + 60x + 35 + 60x + 35
Inverse FunctionsInverse Functions
Given:Given: f(x) = 2xf(x) = 2x g(x) = x/2g(x) = x/2
Note:Note: f(g(x)) = f(x/2) = 2(x/2) = xf(g(x)) = f(x/2) = 2(x/2) = x g(f(x)) = g(2x) = (2x)/2 = xg(f(x)) = g(2x) = (2x)/2 = x
f(x) “undoes” the effect of g(x) and f(x) “undoes” the effect of g(x) and g(x) “undoes” the effect of f(x)g(x) “undoes” the effect of f(x)
f is the inverse of g (f = gf is the inverse of g (f = g-1-1))g is the inverse of f (g = fg is the inverse of f (g = f-1-1))
Inverse FunctionInverse Function
(f(f-1-1 oo f)(x) = f f)(x) = f-1-1(f(x)) = x(f(x)) = x(g(g-1-1 oo g)(x) = g g)(x) = g-1-1(g(x)) = x(g(x)) = x
Given:Given: f(x) = 3x + 2f(x) = 3x + 2 g(x) = (x – 2)/3g(x) = (x – 2)/3
Show that f(x) and g(x) are inverse of each Show that f(x) and g(x) are inverse of each other.other.
Inverse FunctionsInverse Functions
Given:Given: f(x) = 3x + 2f(x) = 3x + 2 g(x) = (x – 2)/3g(x) = (x – 2)/3
f(g(x)) = f((x – 2)/3)f(g(x)) = f((x – 2)/3) = 3((x – 2)/3) + 2 = 3((x – 2)/3) + 2 = x – 2 + 2 = x = x – 2 + 2 = x
g(f(x)) = g(3x + 2)g(f(x)) = g(3x + 2) = ((3x + 2) – 2)/3 = ((3x + 2) – 2)/3 = (3x + 2 – 2)/3 = (3x + 2 – 2)/3 = 3x/3 = 3x/3 = x = x
Finding the Inverse of a FunctionFinding the Inverse of a Function
Given: f(x) = 7x – 5Given: f(x) = 7x – 5 Find: fFind: f-1-1(x)(x)
Let f(x) = yLet f(x) = yy = 7x – 5y = 7x – 5
Interchange x and yInterchange x and yx = 7y – 5x = 7y – 5
Solve for ySolve for y(x + 5)/7 = y(x + 5)/7 = y
Replace y with fReplace y with f-1-1(x)(x) f f-1-1(x) = (x + 5)/7(x) = (x + 5)/7
Finding the Inverse of a FunctionFinding the Inverse of a Function Given: f(x) = xGiven: f(x) = x33 + 1 + 1 Find: fFind: f-1-1(x)(x)
Let f(x) = yLet f(x) = yy = xy = x33 + 1 + 1
Interchange x and yInterchange x and yx = yx = y33 + 1 + 1
Solve for ySolve for yx – 1 = yx – 1 = y33
(x – 1)(x – 1)1/3 1/3 = y= y Replace y with fReplace y with f-1-1(x)(x)
f f-1-1(x) = (x – 1)(x) = (x – 1)1/3 1/3
9.3 Logarithmic Function9.3 Logarithmic Function
Alaska Earthquake (1964, 131 killed)Alaska Earthquake (1964, 131 killed) Magnitude: 9.1Magnitude: 9.1
Hawaii Earthquake (1951)Hawaii Earthquake (1951) Magnitude: 6.9Magnitude: 6.9
Chile Earthquake (1960)Chile Earthquake (1960) Magnitude: 9.5Magnitude: 9.5
How many times is the energy released by How many times is the energy released by earthquake of magnitude of 9 compared to 7?earthquake of magnitude of 9 compared to 7?
Inverse of f(x) = bInverse of f(x) = bxx
Given exponential function f(x) = bGiven exponential function f(x) = bxx
Wha is the inverse of f(x), i.e., fWha is the inverse of f(x), i.e., f-1-1(x)?(x)? To find inverse:To find inverse:
Let f(x) = yLet f(x) = y
y = by = bxx
Exchange x & yExchange x & y
x = bx = byy
Solve for ySolve for yy = ?y = ?
y = logy = logbbxx (This is a new notation.) (This is a new notation.)
(log(logbbx = x = exponentexponent to base b such that b to base b such that byy = x) = x)
Equivalence of Exponential Form Equivalence of Exponential Form and Logarithmic Formand Logarithmic Form
LogLog55x = 2 means x = 5x = 2 means x = 522
loglog4426 = y means 426 = y means 4yy = 26 = 26
121222 = x means log = x means log1212x = 2x = 2
ee66 = 33 means y = log = 33 means y = logee3333 Remember, logarithm (of a number) means Remember, logarithm (of a number) means
exponent (of a number)exponent (of a number)
Evaluating LogarithmEvaluating Logarithm
x = logx = log1010100 means 10100 means 10xx = 100 = 100
Thus, x = 2 Thus, x = 2 y = logy = log36366 means 366 means 36yy = 6 = 6
Thus, y = 0.5Thus, y = 0.5 z = logz = log228 means 28 means 2zz = 8 = 8
Thus, y = 3Thus, y = 3 x = logx = log777788 means 7 means 7xx = 7 = 788
Thus, x = 8Thus, x = 8
Your TurnYour Turn Solve for x.Solve for x.
1.1. x = logx = log55125125 55x x = 125= 125
= 5 = 533
Thus, x = 3Thus, x = 3
2.2. x = 3x = 3loglog33
1717
Let y = logLet y = log331717
x = 3x = 3yy
loglog33x = yx = y
loglog33x = logx = log331717
Thus, x = 17Thus, x = 17
Graph of Logarithmic FunctionGraph of Logarithmic Functionf(x) = 2x g(x) = logx2
x f(x) x g(x)
-2 0.25 0.25 -2
-1 0.5 0.5 -1
0 1 1 0
1 2 2 1
2 4 4 2
3 8 8 3
f(x) = 2x
g(x) = log2x
y = x
(1,0)
(0,1)
x
y
Domain & Range of bDomain & Range of bxx and log and logbbxx
f(x) = 2x
g(x) = log2x
y = x
(1,0)
(0,1)
x
yf(x) = bx
•Domain: (-∞, ∞)
•Range: (0, ∞)
g(x) = logbx
•Doman: (0, ∞)
•Range: (-∞, ∞)
Common LogarithmCommon Logarithm
Common log of a number—to base 10.Common log of a number—to base 10. loglog1010100 = log100 = 2100 = log100 = 2
loglog10101000 = log 1000 = 31000 = log 1000 = 3
loglog10100.01 = log 0.01 = -20.01 = log 0.01 = -2
Richter ScaleRichter Scale II
R = log ----- where IR = log ----- where I00 is the intensity of is the intensity of
I I0 0 barely felt 0-level earthquakebarely felt 0-level earthquake
RRAA = log(I = log(IAA/I/I00) => 10) => 10RRAA = I = IAA/I/I00
RRHH = log(I = log(IHH/I/I00) => 10) => 10RRHH = I = IHH/I/I00
1010RRA A (I(IAA/I/I00))
------ = --------------- = ---------1010RR
HH (I (IHH/I/I00))
101099/10/1077 = I = IAA/I/IHH
== 101022 = 100= 100
9.4 Propertis of Logarithms 9.4 Propertis of Logarithms
Product RuleProduct Rule Recall: bRecall: bmm ∙∙ b bnn = b = bmm + n+ n
loglogbb(b(bmm ∙∙ b bnn) = m + n) = m + n Thus, for M, N > 0, b Thus, for M, N > 0, b ≠ ≠ 1:1:
loglogbb(M (M ∙∙ N) = log N) = logbbM + logM + logbbN N
Quotient RuleQuotient Rule For M, N > 0, b For M, N > 0, b ≠ ≠ 1:1:
loglogbb(M (M / / N) = logN) = logbbM - logM - logbbN N
Power RulePower Rule For M > 0, b For M > 0, b ≠ ≠ 1, and p 1, and p εε RR
loglogbb(M(Mpp) = p ) = p ∙∙ log logbbMM
Using Properties of LogarithmsUsing Properties of Logarithms
Expand the following:Expand the following: log(10x)log(10x)
= log 10 + log x = log 10 + log x = 1 + log x= 1 + log x
loglog22 (8/x) (8/x) loglog228 – log8 – log22x x
= 3 - log= 3 - log22xx loglog55 7 744
= 4 = 4 ∙∙ log log5577 log log √(√(x)x)
= (0.5) = (0.5) ∙ log x∙ log x
9.5 Exponential and Logarithmic 9.5 Exponential and Logarithmic EquationsEquations
Exponential EquationExponential Equation Equation containing variable in exponentEquation containing variable in exponent
ExamplesExamples 2323x-8x-8 = 16 = 16 44xx = 15 = 15 40e40e0.6x0.6x = 240 = 240
Solving Exponential EquationSolving Exponential Equation
If bIf bMM = b = bNN, then M = N , then M = N Solve: 2Solve: 23x-83x-8 = 16 = 16
223x-83x-8 = 2 = 244
3x – 8 = 43x – 8 = 43x = 123x = 12x = 4x = 4
1616xx = 64 = 64 (4(422))xx = 4 = 433
442x2x = 4 = 433
2x = 32x = 3x = 3/2x = 3/2
Solving Exponential EquationSolving Exponential Equation SolveSolve
55xx = 134 = 134 log (5log (5xx) = log (134)) = log (134)
x log 5 = log 134x log 5 = log 134x = log 134 / log 5x = log 134 / log 5 ≈≈ 2.127/0.699 2.127/0.699 ≈≈ 3.043 3.043
Check: 5Check: 53.0433.043 ≈≈ 134 134
1010xx = 120,000 = 120,000 log(10log(10xx) = log(120,000)) = log(120,000)
x = log (120,000)x = log (120,000) = log(1.2 = log(1.2 ∙∙ 100000) 100000) = log 1.2 = log 1.2 + 5 + 5 ≈ ≈ 0.079 + 5 0.079 + 5 ≈≈ 5.079 5.079
Check: 10Check: 105.0795.079 ≈≈ 120,000 120,000
Logarithmic EquationLogarithmic Equation
Solve:Solve: loglog22x + logx + log22(x – 7) = 3(x – 7) = 3
loglog22(x (x ·· (x – 7)) = 3 (x – 7)) = 3x(x – 7) = 2x(x – 7) = 233
xx22 – 7x = 8 – 7x = 8xx22 – 7x – 8 = 0 – 7x – 8 = 0(x + 1)(x – 8) = 0(x + 1)(x – 8) = 0x = -1, 8x = -1, 8
Check: Check: for x = 8 for x = -1for x = 8 for x = -1
loglog228 + log8 + log22(8 – 7) = 3 ? log(8 – 7) = 3 ? log2 2 (-1) + log(-1) + log22(-8) = 3 ? (-8) = 3 ? 3 + 0 = 3 Yes No. log of negative 3 + 0 = 3 Yes No. log of negative undefinedundefined
Your TurnYour Turn 55xx = 17 = 17
log(5log(5xx) = log(17)) = log(17)x log 5 = log 17x log 5 = log 17x = log 17 / log 5x = log 17 / log 5 ≈ 1.230 / 0.699≈ 1.230 / 0.699 ≈ 1.761≈ 1.761
Check: 5Check: 51.7611.761 ≈ 17 ≈ 17 loglog33(x + 4) = log(x + 4) = log3377
33xx – 4– 4 = 3 = 377
x – 4 = 7x – 4 = 7x = 11x = 11
Check: 3Check: 311 - 411 - 4 = 3 = 377
Application (skip)Application (skip)
The percentage of surface sunlight, f(x), that The percentage of surface sunlight, f(x), that reaches a depth of x feet beneath of the surface reaches a depth of x feet beneath of the surface of the ocean is modeled by:of the ocean is modeled by:
f(x) = 20(0.975)f(x) = 20(0.975)xx
Calculate at what depth there is 1% of surface Calculate at what depth there is 1% of surface sunlight.sunlight.
f(x) = 20(0.975)f(x) = 20(0.975)xx
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
20 40 60 80 100 120 140
Depth
% o
f S
urf
ac
e S
un
ligh
t
f(x) = 20(0.975)f(x) = 20(0.975)xx
1 = 20(0.975)1 = 20(0.975)xx
0.05 = 0.9750.05 = 0.975xx means meansloglog0.9750.975 0.05 = x (calculater has no log 0.05 = x (calculater has no log0.9750.975))
lnlnee 0.05 = ln 0.05 = lnee0.9750.975xx
= x = x ∙ ∙ lnlnee0.9750.975 ln 0.05 / ln 0.975 = xln 0.05 / ln 0.975 = xx = -2.996/-0.025 x = -2.996/-0.025 ≈ 118 (feet)≈ 118 (feet)
ApplicationApplication
The function The function
P(x) = 95 – 30 logP(x) = 95 – 30 log22x x
models the percentage P of students who could recall models the percentage P of students who could recall the important features of a classroom lecture as a the important features of a classroom lecture as a function of time (x is number of days)function of time (x is number of days)
After how many days do only half the students recall After how many days do only half the students recall the important features of a classroom lecture?the important features of a classroom lecture?
P(x) = 95 – 30 logP(x) = 95 – 30 log22x x
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20.00
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100.00
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Days
%
SolutionSolution
P(x) = 95 – 30logP(x) = 95 – 30log22xx
50 = 95 – 30log50 = 95 – 30log22xx
30log30log22x = 95 – 50x = 95 – 50
loglog22x = 45/30x = 45/30
loglog22x = 1.5 meansx = 1.5 means
x = 2x = 21.51.5
≈ 2.8 (days)≈ 2.8 (days)