Chapt. 9Chapt. 9Exponential and Exponential and

Logarithmic FunctionsLogarithmic Functions

Mall Browsing Time vsMall Browsing Time vsAverage Amount SpentAverage Amount Spent

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$50.00

$100.00

$150.00

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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

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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

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5.00

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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

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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))

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5.00

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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

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-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

0.00

20.00

40.00

60.00

80.00

100.00

1 2 3 4 5 6 7 8 9 10

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)