Exponential and logarithmic functions - WordPress.com · Exponential and logarithmic functions....

transcript

.. الطالبات

خديجه , مالك المسعد , نـدى الغـمـيـــز , حصه الرومي

الزهراني

..األستاذة

يــســــــرى

Exponential and logarithmic functions

Modeling with Exponential and Logarithmic Functions

Exponential Growth and Decay Models

The mathematical model for exponential growth or decay is given by

f (t) = A0ekt or A = A0e

•If k > 0, the function models the amount or size of a growing entity. A0

is the original amount or size of the growing entity at time t = 0. A is the amount at time t, and k is a constant representing the growth rate.

•If k < 0, the function models the amount or size of a decaying entity. A0 is the original amount or size of the decaying entity at time t = 0. A is

the amount at time t, and k is a constant representing the decay rate.

The graph below shows the growth of the Mexico City metropolitan area

from 1970 through 2000. In 1970, the population of Mexico City was 9.4

million. By 1990, it had grown to 20.2 million.

•Find the exponential growth function that models the data.

•By what year will the population reach 40 million?

1970 1980 1990 2000

Example

a. We use the exponential growth model

A = A0ekt

in which t is the number of years since 1970. This means that 1970

corresponds to t = 0. At that time there were 9.4 million inhabitants, so we

substitute 9.4 for A0 in the growth model.

A = 9.4 ekt

We are given that there were 20.2 million inhabitants in 1990. Because

1990 is 20 years after 1970, when t = 20 the value of A is 20.2. Substituting

these numbers into the growth model will enable us to find k, the growth

rate. We know that k > 0 because the problem involves growth.

A = 9.4 ektUse the growth model with A0 = 9.4.

20.2 = 9.4 ek•20When t = 20, A = 20.2. Substitute these values.

Example cont

We substitute 0.038 for k in the growth model to obtain the exponential

growth function for Mexico City. It is A = 9.4 e0.038t where t is measured in

years since 1970.

Example cont

20.2/ 9.4 = ek•20Isolate the exponential factor by dividing both sides by 9.4.

ln(20.2/ 9.4) = lnek•20Take the natural logarithm on both sides.

0.038 = kDivide both sides by 20 and solve for k.

20.2/ 9.4 = 20kSimplify the right side by using ln ex = x.

b. To find the year in which the population will grow to 40 million, we

substitute 40 in for A in the model from part (a) and solve for t.

Because 38 is the number of years after 1970, the model indicates that the

population of Mexico City will reach 40 million by 2008 (1970 + 38).

A = 9.4 e0.038t This is the model from part (a).

40 = 9.4 e0.038t Substitute 40 for A.

Example cont

ln(40/9.4) = lne0.038t Take the natural logarithm on both sides.

ln(40/9.4)/0.038 =t Solve for t by dividing both sides by 0.038

ln(40/9.4) =0.038t Simplify the right side by using ln ex = x.

40/9.4 = e0.038t Divide both sides by 9.4.

•Use the fact that after 5715 years a given amount of carbon-14 will have

decayed to half the original amount to find the exponential decay model

for carbon-14.

•In 1947, earthenware jars containing what are known as the Dead Sea

Scrolls were found by an Arab Bedouin herdsman. Analysis indicated

that the scroll wrappings contained 76% of their original carbon-14.

Estimate the age of the Dead Sea Scrolls.

Solution

We begin with the exponential decay model A = A0ekt. We know that k < 0

because the problem involves the decay of carbon-14. After 5715 years

(t = 5715), the amount of carbon-14 present, A, is half of the original

amount A0. Thus we can substitute A0/2 for A in the exponential decay

model. This will enable us to find k, the decay rate.

Text Example

Substituting for k in the decay model, the model for carbon-14 is

A = A0e–0.000121t.

k = ln(1/2)/5715=-0.000121Solve for k.

1/2= ekt5715Divide both sides of the equation by A0.

Solution

A0/2= A0ek5715After 5715 years, A = A0/2

ln(1/2) = ln ek5715Take the natural logarithm on both sides.

ln(1/2) = 5715kln ex = x.

Text Example cont

Exponential Functions

Exponential FunctionThe w/ base a for a > 0

* For a 1,

- Domain: (–, )

- Range: (0, )

for a > 1 xaxf

for 0 < a < 1 xaxf

Natural Exponential Function

with base e

Compound Interest

-A(t) = amount after t years-P = Principal-r = interest rate-n = # of times interest is compounded per year-t = number of years

nrPtA )1(

Continually Compounded Interest

-A(t) = amount after t years

-P = Principal

-r = interest rate

-t = number of years

rtPetA

If $350,000 is invested at a rate of 5½% per year, find the amount of the investment at the end of 10 years for

the following compounding methods:

Quarterly -

- Monthly

- Continuously

Exponential Growth

n(t) = population at time t

no = initial size of population

r = rate of growth

t = time

rtoentn

Logarithmic Functions

The Log Function is the inverse of the Exponential Function, so…

f x ax

f1x loga x

And… for

Domain: (0, )

Range: (–, )

f x loga x

a is positive with a 1

exponential form

logarithmic form

base exponent base exponent

Common Logarithm(base 10)

xx 10loglog

Natural Logarithm(base e)

xx elogln

xeyx y ln

Properties of Logs

01log a

1log aa

xaxa log

Properties of Natural Logs

01ln 1ln e

xex ln

xe x ln

Exponential and Logarithmic Functions

These functions model rapid growth or decay:

- # of users on the Internet

16 million (1995) 957 million (late 2005)

- Compound interest

- Population growth or decline

Comparison

- Linear Functions

Rate of change is constant

- Exponential Functions

Change at a constant percent rate.

The Exponential Function

y = abx

b is the base:

- It must be greater than 0

- It cannot equal 1.

x can be any real number

Identify Exponential Functions

* Which of the following are exponential functions?

y = 3x y = x3

y = 2(7)x y = 2(-7)x

Identify the Base

* Identify the base in each of the following.

y = 3x y = 2(7)x

y = 3axy = 4x - 3

Evaluate Exponential Functions

y = 3x for x = 4

y = 2(7)x for x = 3

y = -2(4x) for x = 3/2

Graph Exponential Functions (b > 1)

* Graph y = 2x for x = -3 to 3

Graph of y = 2^x

-4 -3 -2 -1 0 1 2 3 4

Graph Exponential Functions (0< b < 1)

* Graph y = (1/2)x for x = -3 to 3

Graph of y = (1/2)^x

-4 -3 -2 -1 0 1 2 3 4

Summary

y = abx

- x can be any value

- The resulting y value will always be positive.

- The y-intercept is always (0,1)

- When b > 1, as x increases, y increases.

- When 0 < b < 1, as x increases, y decreases.

Practice

* Using Microsoft Excel:

- Graph the function y = 3x for x = -3 to 3

(in 0.5 increments)

- Graph the function y = (1/3)x for x = -3 to 3

(in 0.5 increments)

Practice: Graph using Excel

- Result:Graphing Exponential Functions

-3 -2 -1 0 1 2 3

Suppose you are a salaried employee, that is, you are paid afixed sum each pay period no matter how many hours youwork. Moreover, suppose your union contract guaranteesyou a 5% cost-of-living raise each year. Then your annual

salary is an increasing function of the number of years youhave been employed, because your annual salary will

increase by some amount each year. However, theamount of the increase is different from year to year,

because as your salary increases, the amount of your 5%raise increases too. This phenomenon is known as

compounding.

Example

Assume your starting salary is $28,000 per year. Let S(t) be your annual salaryafter full years of employment. Therefore, S(0) is interpreted to mean your

initial salary of $28,000. How can we evaluate S(1), your salary after 1 year ofemployment? Since your salary is increasing by 5% each year, this means

S(1) is 5% more than S(0). In other words, S(1) is 105% of S(0). Thus, we canevaluate S(1) as shown here, by changing the percentage 105% to a decimal

number:S(1) = 105% of S(0) = 1.05 × S(0) = 1.05 × 28000S(2) = 105% of S(1) = 1.05 × S(1) = 1.052 × 28000S(3) = 105% of S(2) = 1.05 × S(2) = 1.053 × 28000S(4) = 105% of S(3) = 1.05 × S(3) = 1.054 × 28000S(5) = 105% of S(4) = 1.05 × S(4) = 1.055 × 28000

Example

Graph of Exponential Functions

Exponential functions have symbol rules of the formf (x) = c ⋅bx

b: base or growth factor -- must be positive real number but cannot be 1, i.e. b > 0 and b ≠ 1

c: coefficient greater than 0the domain of f is (−∞, ∞)the range of f is (0, ∞)

Natural Exponential Function

f (x) = ex f (x) = e− x

Example

Find the exponential function whosegraph as shown below.

Consider the exponential function f shown here withbase b = 2 and initial value c = 1

Suppose we want to find the input number for thatmatches the output values 8 and 15, in other

words, we want to solve the equation

Let's introduce a new function designed to help usexpress solutions to equations like the two shownhere, which are solved by finding particular inputnumbers for the exponential function f. We give

this new function a special label:

helps us express inputs for the function f. Thus,for example, we evaluate , because f(3)=

. Likewise, we evaluate

In general,

That is exponential function and logarithmic function are inverseof each other.

Common and Natural Logarithms

• A common logarithm is a logarithm with base 10,log10.

• A natural logarithm is a logarithm with base e, ln.

Properties of Logarithms

Graphs of Logarithmic Functions

Laws of Logarithms

Change of Base

Compound Interest

If P is a principal of an investment with an interest rfor a period of t years, then the amount A of theinvestment is

Modeling with Exponential andLogarithmic Functions

Exponential Growth Model

A population that experiences exponential growthincreases according to the model

population at time tinitial size of populationrelative rate of growthtime

Radioactive Decay Model

If m0 is the initial mass of a radioactive substancewith half-life h, then the remaining mass ofradioactive at time t is modeled by

Total differential of Q using logs

LdKdAdQd

lnlnlnln

Exponent example

K and Lfor Solve

38)-337 pp. 5, (Example

firm a of decisionsInput 11.6(c)

P-rKLαπ

P-wKLαπ

-wL-rKKLPπ

.αKLQ

PQ-wL-rKR-Cπ

Maximization conditions

years 251004

%10)(let

bottle/38.148$

bottle/18.12$

bottle/1$)(let

keA(t)

))(((.

10.6(c) Timber cutting problemplot of .5(t)-.5ln(2)=r, r=.05, t=48

10.7(b) Rate of growth of a combination of

functions; Example 3 consumption & pop.

vvuuvuvu

rsrsrvu

vrtgurtftf

tgtfvu

tgtfdt

tgvtfuvuz

)('and,)('then,)(

))(')('(1

))()((1

)()ln(

)ln()ln(

Trigonometric, Logarithmic,

and Exponential Functions

In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those

properties which will be useful in future math and science applications.

Trigonometric Functions

Geometrically, there are two ways to describe trigonometric functions:

Trigonometric FunctionsGeometrically, there are two ways to describe trigonometric

functions: Polar Anglex=cos y=sin Measure in radians: =radiusarc length For

example,180=rr= radians Radians=180degrees

Graph Exponential Functions(0< b < 1)

Graph y = (1/2)x for x = -3 to 3

Graph of y = (1/2)^x

-4 -3 -2 -1 0 1 2 3 4

Summaryy = abx

•x can be any value

•The resulting y value will always be

positive.

•The y-intercept is always (0,1)

•When b > 1, as x increases, y

increases.

•When 0 < b < 1, as x increases, y

decreases.

Substituting for k in the decay model, the model for carbon-14 is

A = A0e–0.000121t.

k = ln(1/2)/5715=-0.000121 Solve for k.

1/2= ekt5715 Divide both sides of the equation by A0.

Solution

A0/2= A0ek5715 After 5715 years, A = A0/2

ln(1/2) = ln ek5715 Take the natural logarithm on both sides.

ln(1/2) = 5715k ln ex = x.

Text Example cont

Solution

The Dead Sea Scrolls are approximately 2268 years old plus the number of

years between 1947 and the current year.

A = A0e-0.000121t This is the decay model for carbon-14.

0.76A0 = A0e-0.000121t A = .76A0 since 76% of the initial amount remains.

0.76 = e-0.000121t Divide both sides of the equation by A0.

ln 0.76 = ln e-0.000121t Take the natural logarithm on both sides.

ln 0.76 = -0.000121t ln ex = x.

Text Example cont.

t=ln(0.76)/(-0.000121) Solver for t.

Finally we hope you can

understand the logarithmic

function , and we hope you like

our project =)

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