exponential and logarithmic functions. exponential functions example: graph the following...

Exponential and Logarithmic Functions

Exponential Functions

Example: Graph the following equations…

a)

b)

xy 2

x

y3

1

An exponential function is a function of the form

fx ax()

where a is a positive real number (a > 0) and a 1. The domain of f is the set of all real numbers.

Domain: All real numbersRange: (0, )No x-interceptsy-intercept: 1Horizontal asymptote: y = 0 as x -Increasing functionOne-to-one

Summary of the characteristics of the graph

f x a ax( ) , 1

Summary of the characteristics of the graph

f x a ax( ) , 0 1

Domain: All real numbersRange: (0, )No x-interceptsy-intercept: 1Horizontal asymptote: y = 0 as x Decreasing functionOne-to-one

Graph and determine

the domain, range and horizontal

asymptote of

f x

f

x( )

.

3 2

0

5

10

(0, 1)

(1, 3)

y x3

0

5

10

(0, 1)(-1, 3)

y x 3

0

5

10

(0, 3)

(-1, 5)

y = 2

y x 3 2

T h e n u m b e r e i s d e f in e d a s th e n u m b e r th a t th e e x p r e s s io n

11

n

n

a p p r o a c h e s a s n . T h is i s e x p r e s s e d u s in g l im i t n o ta t io n a s

enn

n

l im 11

e 2 718281827.

3 2 1 0 1 2 3

2

4

6

y x2

y x3 y ex

Horizontal Asymptote: y = 2

Range: { y | y >2 } or (2,

Domain: All real numbers

)

The logarithm function to the base a, where

a > 0 and a 1, is denoted by y xalog (readas “y is the logarithm to the base a of x”) and isdefined by

y x x aay log if and only if

If then y x xy log ,3 3

If then b b3 8 3 8 , log

Properties of the Logarithm Function

implies

Domain: Range:

y x x a

x ya

y

log

0

Exponential Growth or Decay

The amount N after t time periods (usually years) due to an initial amount , where r is the rate of growth (or decay) can model exponential growth (or decay) with the following formula:

trNN )1(0

0N

Interest is the money paid for the use of money. The total amount borrowed is called the principal. The rate of interest, expressed as a percent, is the amount charged for the use of the principal for a given period of time, usually on a yearly (per annum) basis.

Simple Interest Formula

I = Prt

Compound Interest Formula

The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is

APrn

nt

1

(a) Annually

A Prn

nt

1 )3(1

1

04.01500$

$562.43

(b) Monthly

A Prn

nt

1

$500

. ( )

10 0412

12 3

$563.64

Suppose your bank pays 4% interest per year. If $500 is deposited, how much will you have after 3 years if interest is compounded …

Formula: Continuous Compounding Interest

The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is

APert

Suppose your bank pays 4% interest per year. If $500 is deposited, how much will you have after 3 years if interest is compounded continuously?

A Pert $500 . ( )e0 04 3

$563.75

Compound Interest Problem

A Goofy Situation

Goofy made purchases on his Magic Kingdom Mastercard totaling $1200. The interest is compounded monthly on his card, at a rate of 19.5%.

If Goofy forgets to pay of his credit card balance (as he usually does), how much will he owe after 2 years?


Mickey Mouse puts $400 into the bank. He chose the EPCOT Savings and Loan, which is owned by Scrooge McDuck, where interest is compounded continuously.

What interest rate was offered by Scrooge, if Mickey had $432 in his account after 6 years?


Donald Duck invests his money in MGM Studios Credit Union. The Credit Union compounds interest daily, at a rate of 2%.

How long will it take Donald to double his money?

N Ne kt 0

Formula: Exponential Growth and Decay (involving ‘e’)

Present and Future Value of an AnnuityAn Annuity is a series of equal payments made at equal intervals of time. The present value of an annuity, , is the sum of the present values of all the periodic payments (P).

The lump-sum investment of dollars now will provide payments of P dollars for n periods.

This formula can be used to find the present value, where i is for interest, n is for the number of payments

i

iPP

n

n

)1(1

nP

nP

Present and Future Value of an Annuity

The future value of an annuity, , is the sum of all of the annuity payments plus any accumulated interest.

This formula can be used to find the future value, where i is for interest, n is for the number of payments

Note: APR stands for “Annual Percentage Rate”

i

iPF

n

n

1)1(

nF


Example: What is the present value of an annuity that pays out $2500 every week for 3 years if the APR is 6%?


Example: What are the monthly payments on a $11,000 car over 6 years with an APR of 10%?


Example: In order to have $50,000 at the end of 10 years, what amount should you deposit each month into an account with an APR of 7.5%?


Example: How much money would the state of Pennsylvania need to have in an account with an APR of 11% in order to pay out $500,000 to a state Lotto winner in equal annual payments over the next 15 years?

0

y a x

y xalog

y x

(0, 1)

(1, 0)

0 < a < 1

0

y a x

y xalog

y x

(1, 0)

(0, 1)

a > 1

Facts about the Graph of a Logarithmic

Function f x xa( ) log

1. The x-intercept of the graph is 1. There is no y-intercept.

2. The y-axis is a vertical asymptote of the graph.

3. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1.

4. The graph is smooth and continuous, with no corners or gaps.

y x x ey ln if and only if

If th e n a a u vu v ,

Evaluate: log4 64

y log4 64

4 64y

4 43y y 3

27log9

Simplify the following expressions:

4

1ln

e

5)4ln( x

Solve the following equations:

4log3 x

)6(log2

log log loga a aMN M N

log log loga a aMN

M N

log logar

aM r M

Properties of Logarithms

Write the following expression as the sum

and / or difference of logarithms. Express

all powers as factors.

logax x

x

2

2

3

1

log log loga a a

x x

xx x x

2

22 23

13 1

log log loga a ax x x2 23 1

212

3 2 1log log loga a ax x x

Write the expression

as a single logarithm.

314


314


log log loga a ax x x3 142 1 1

log log loga a ax x x3 4 2 1 1

log loga ax x x3 4 2 1 1

loga

x xx

3 4 2 11

Solve: log4 3 2 2x

log4 3 2 2x

3 2 42x

3 2 16x 3 18x x 6

Solve: log log6 63 2 1x x log log6 63 2 1x x

log6 3 2 1x x

log62 6 1x x

x x2 16 6 x x2 12 0 x x 4 3 0

x x 4 3 or Solution set: {x | x = 3}

Solve: 9 3 10 02x x

9 3 10 02x x

3 3 3 10 02 2x x

3 9 3 10 02x x

3 10 3 1 0x x

3 10 0 3 1 0x x or

3 10 3 1x x or

No Solution x 0

Solution set: x x 0

Solve: 4 642x

4 642x

4 42 3x x 2 3

x 5

Solve: 7 53 2x x

7 53 2x x

ln ln7 53 2x x

x xln ln7 3 2 5 x xln ln ln7 3 5 2 5

x xln ln ln7 3 5 2 5 x ln ln ln7 3 5 2 5

x

2 5

7 3 5

ln

ln ln1117.

Logarithms and Antilogs

Logarithms with base 10 are called Common Logarithms.

Antilogarithm: if log x = a, then x = antilog a.

Or, you can say that x = 10^a.

Logarithms and Antilogs

Logarithms with base ‘e’ are called Natural Logarithms and are written as ‘ln’

Anti-natural logarithm: if ln x = a,

then x = antiln a. Or, you can say that x = e^a.

Change in Base Formula

If a, b, and n are positive numbers and neither a nor b is 1, then the following formula holds true:

a

nn

b

ba log

loglog

Domain of logarithmic function = Range ofexponential function = (0, )

Range of logarithmic function = Domain ofexponential function = (-, )

A Prn

nt

1

2 1010

4

4

P Pt

.

2 1010

4

4

. t

ln ln.

2 1010

4

4

t

ln ln.

2 4 1010

4

t

t

ln

ln.

2

4 1010

4

7 018. years

How long will it take to double an investment earning 10% per year compounded quarterly?

rtePA

tePP 10.2

te 10.2

te 10.ln2ln

et ln10.2ln

10.

2lnt years 931.6

How long will it take to double an investment earning 10% per year compounded continuously?

Uninhibited Growth and Decay Formula

ktney

Where n is the initial amount, y is the final amount, t is for time, and k is a constant (depending on the substance that is growing or decaying).

N t Ne kt( ) 0

k > 0

Uninhibited Growth (cells, bacteria, etc.)

N Ne kt 0

Exponential Growth or Decay Formula

(in terms of e)

for growth, k > 0

for decay, k < 0

N N0 100 1 175 ( )

N t N ekt( ) 0

N ek( ) ( )1 175 100 1

175. ek

ln .175 kk 0 5596.

A culture of bacteria increases according to the law of uninhibited growth. If 100 grams bacteria are present initially, and there are 175 grams of bacteria after 1 hour, how many will be present after 4 hours?

k 0 5596.

N t e t( ) .100 0 5596

N e( ) . ( )4 100 0 5596 4937 8. grams

N t N ekt( ) 0

Uninhibited Growth

Example: For a certain strain of bacteria, k = .0325 and tis measured in days. How long will it take 25 bacteria to increase to 500?

A t Ae kt( ) 0

k < 0

Uninhibited Decay

Uninhibited Radioactive Decay

Example: The half-life of a certain radioactive substance is500 years. If 300 kg of the substance is present now, how much will be present in 800 years?

The half-life of Uranium-234 is 200,000 years. If 50 grams of Uranium-234 are present now, how much will be present in 1000 years. NOTE: The half-life is the time required for half of the radioactive substance to decay.

A t A ekt( ) 0

A t A( ) 12 0

12 0 0A A ekt

12

200 000ek ( , )

ln ,12

200 000

k

k

ln

,

12

200 000 3 466 10 6.

A t A e t( ) .

03 466 10 6

A e( ) . ( )1000 50 3 466 10 10006

49 8. grams

k 3 466 10 6.

A t A ekt( ) 0

Uninhibited Radioactive Decay

Example: If 18 grams of a certain radioactive substancedecays to 5 grams after 340 years, what is its half-life?

The half-life of a certain radioactive substance is 2500 years. If 35 grams of the substance are present now, how much will be present in 1500 years?

Newton’s Law of Cooling

u t T u T e kkt( ) 0 0

T : Temperature of surrounding medium

uo : Initial temperature of object

k : A negative constant

A cup of hot chocolate is 100 degrees Celsius. It is allowed to cool in a room whose air temperature is 22 degrees Celsius. If the temperature of the hot chocolate is 85 degrees Celsius after 4 minutes, when will its temperature be 60 degrees Celsius?

u t T u T ekt( ) 0 85 22 100 22 4 ek ( )

63 78 4 e k

6378

4e k

ln6378

4

k

u t e t( ) . 22 78 0 0534

k

ln

63784 60 22 78 0 0534 e t.

38 78 0 0534 e t.

3878

0 0534 e t.

ln .1939

0 0534

t

t

ln

.

1939

0 0534135. minutes

0 0534.

u t T u T ekt( ) 0

T u 22 1000,


y : temperature of cooling object

c: Temperature of surrounding medium

a: initial temp – temp of surrounding medium

k : a constant

t : time it takes for object to cool

caey kt


Example: Find the time it takes a cup of soup to cool to 100 degrees, if it has been heated to 180 degrees. Room temperature is 70 degrees and k = .01.

Newton’s Law of Cooling: Time of Death

At 4:45 AM, the CSI Team was called to the home of a person who had died during the night. In order to estimate the time of death, Grissom took the person’s body temperature twice. At 5:00 AM , the body temp was 85.7 F and at 5:30, the temp was 82.8 F. The room temperature was 70 . Find the approximate time of death.

Logarithmic Application: pH level of a chemical solution based on the hydrogen ions present

pH = -log10[H+]

Example: The pH of a solution is 5.3. Find the hydrogen ion concentration.

Logarithmic Application: pH level of a chemical solution based on the hydrogen ions present

pH = -log10[H+]

Example: Find the pH level for a certain solution with a hydrogen ion concentration of .00024

Logarithmic Application: Decibel Level of Sound (D) based on the intensity (I) of its sound waves ( in watts per square meter)

1210 10

log10I

D

Example: Find the number of Decibels if the intensity of the sound generates 10-7 watts per square meter at a distance of 12 feet.

Logarithmic Application: Decibel Level of Sound (D) based on the intensity (I) of its sound waves ( in watts per square meter)

1210 10

log10I

D

Example: At a recent concert, the Decibel level registered at 95 decibels. Find the intensity in watts per square meter of the sound.

Logarithmic Application: Earthquake Magnitude (M) on the Richter Scale based on seismographic reading in millimeters (x) divided by a zero-level earthquake ( )

010log

x

xM

Example: Find the Richter Scale reading for an earthquake 75,342 millimeters, 100 kilometers from the center of the quake.

30 10x

Logarithmic Application: Earthquake Magnitude (M) on the Richter Scale based on seismographic reading in millimeters (x) divided by a zero-level earthquake ( )

010log

x

xM

Example: A recent earthquake in California measured 4.1 on the Richter Scale. What would be the magnitude of a quake 100 times stronger?

30 10x

Euler’s Formula

...!6!4!2!0

cos6420

xxxx

x

Leonhard Euler developed the following relationship between trigonometric series and exponential series. Below are the trig series for sine and cosine.

...!7!5!3!1

sin7531

xxxx

x

Euler’s Formula

...!4!3!2!1!0

43210

xxxxx

ex

Here is the exponential series for e^x.

Suppose we replace x with

...!4

)(

!3

)(

!2

)(

!1

)(

!0

)( 43210

iiiii

ei

i

Euler’s Formula

...!4

)(

!3

)(

!2

)(

!1

)(

!0

)( 43210

iiiii

ei

...!4!3!2

1432

iiei

...

!7!5!3...

!6!4!21

753642 iei

Euler’s Formula

sincos iei

...

!7!5!3...

!6!4!21

753642 iei

This can be re-written using the trig series shown earlier to giveus Euler’s Formula:

ireiryix )sin(cos

This formula can be used to write the Exponential Form of aComplex Number, X + Yi…

Euler’s Formula

sincos iei

Now we know that there is no real number that is logarithm of Negative value. However, we can use Euler’s Formula to findA complex number that is the natural log of –1.

sincos iei

1ie

)1ln(ln ie

)1ln(i

Euler’s Formula

Example: Evaluate ln(-12).

Euler’s Formula

)6

sin6

(cos4

i

Example: Write each complex number in Exponential Form.

322 i

i33

Theorem Present Value Formulas

The present value P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year, is

PA

rn

nt

1

If the interest is compounded continuously, then

PAert

How much should you deposit today in order to have $20,000 in three years if you can earn 6% compounded monthly from a bank C.D.?

P Arn

nt

1

$20,. ( )

000 10 0612

12 3

$16, .712 90

Graph

u t e t( ) . 22 78 0 0534

What happens to the value of u(t) as t increases without bound?

Logistic Growth Model

Pt

c

aebt() 1

where a, b, and c are constants with c > 0 and b > 0.

T h e l o g i s t i c g r o w t h m o d e l

P te t( )

. .

5 0 0

1 6 6 7 0 2 4 7 6

r e p r e s e n t s t h e a m o u n t o f a b a c t e r i a ( i ng r a m s ) a f t e r t d a y s .

What is the carrying capacity? 500

Graph the function using a graphing utility.

What was the initial amount of bacteria?

P te t( )

. .

500

1 6 67 0 2476

Pe

( ). . ( )0

500

1 6 67 0 2476 0

500

1 6 67.

65 grams

P te t

( ). .

5001 6 67 0 2476

When will the amount of bacteria be 300 grams?

300500

1 6 67 0 2476

. .e t

1 6 67500300

0 2476 . .e t

6 6753

10 2476. .e t

6 6753

10 2476. .e t

e t

0 2476

53

1

6 67.

.01.

0 2467 01. ln .t

t

ln ..01

0 24679 3. days

Graph Determine the domain,

range, and vertical asymptote.

f x x( ) ln . 3

0 5 10

5

5

(1, 0)

(e, 1)

y xln

0 5 10

5

5

(4, 0)

(e + 3, 1)

y x ln 3

x = 3

Domain: x - 3 > 0x > 3

Range: All real numbers

Vertical Asymptote: x = 3

Between 5:00 p.m. and 6:00 p.m. cars arrive at Wendy’s drive-thru at the rate of 15 cars per hour (0.25 cars per minute). The following formula from statistics can be used to determine the probability a car will arrive within t minutes of 5:00 p.m.

F t e t( ) . 1 0 25

Determine the probability the car will arrive within 5 minutes of 5:00 p.m. (that is, before 5:05 p.m.)

F e( ) . ( )5 1 0 25 5 0 713 713%. .

Determine the probability the car will arrive within 30 minutes of 5:30 p.m. (that is, before 5:30 p.m.)

F e( ) . ( )30 1 0 25 30 0 999 99 9%. .

Graph F(t) using your graphing utility.

What value does F approach as t becomes unbounded in the positive direction?

Growth and Decay in Nature

Final count is y, initial count is , c is the constant of proportionality, t is time, and T is time per cycle of c

T

t

cyy *0

0y

Growth and Decay in Nature

Example: A certain population of single-celled organisms doubles every 3 days. If the initial amount of organisms was 1200 and now there are 100,000, how many days have passed?

loga a 1 a MaMlog loga

ra r

log log loga a aMN M N

log log loga a aMN

M N

log loga aNN

1

log logar

aM r M

Properties of Logarithms

Exponential Functions: Application Problems 1. For a certain radioactive material, the half life is 1200 years.How much of a 400 gram substance would remain after 500years?

2. What are the monthly payments on a mortgage for a $250,000 house over 20 years with APR of 8%?

3. How long would it take for an investment of $300

to reach $1500, if the interest was compounded daily at 6.5%?

Rational Exponents

Example: Express using Rational Exponents

Example: Express using Radicals

3 8627 sr

4

5

4

3

4

1

5 yx

Rational Exponents

Example: Simplify the following…

a)

b)

c)

23484 16* tsrsr

3423 )2(*)4( xyx

2

36 )16( x

exponential and logarithmic functions. exponential functions example: graph the following...

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