Exponential & Logarithmic Functions

•1-to-1 Functions; Inverse Functions•Exponential Functions•Logarithmic Functions•Properties of Logarithms; Exponential & Logarithmic Models•Logarithmic and Exponential Equations•Compound Interest•Applications

{(1, 1), (2, 4), (3, 9), (4, 16)}

one-to-one{(-2, 4), (-1, 1), (0, 0), (1, 1)}

not one-to-one

A function f is said to be one-to-one if, forany choice of numbers x1 and x2, x1 x2, inthe domain of f, then f (x1) f (x2).

One-to-one Functions

Theorem: Horizontal Line Test

If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

500

100

f x( )

2015 x

10 0 10 20

500

Use the graph to determine whether the function f x x x( ) 2 5 12

is one-to-one.

Not one-to-one!

Use the graph to determine whether the function f x x x( ) 3 2 is one-to-one.

f x( )

x

5 0 5

100

100

Is one-to-one!

Let f denote a one-to-one function y f x ( ).

The inverse of f, denoted by f 1, is a

function such that f f x x 1 ( ) for every x

in the domain of f and f f x x 1( ) for

every x in the domain of f 1.

Inverse Functions

Domain of f Range of f

Range of f 1 Domain of f 1

f 1

f

Domain of Range of

Range of Domain of

f f

f f

1

1

Theorem: Graphs of f & f -1 are a reflection image pair

The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

f 1

2 0 2 4 6

2

2

4

6 f

f 1

y = x

(2, 0)

(0, 2)

yx

5

3x

y

5

3

xy x 3 5xy x 3 5

yxx

3 5

f xxx

1 3 5( )

Find the inverse of f xx

x( ) ,

53

The function is one-to-one.

3

Verify and

are inverses.

f xxx

g xx

( ) ( )

3 5 53

f g x fx

( ( ))

53

35

35

53

x

x

35

35

53

33

x

x

xx

35

35

53

33

x

x

xx

15 5 3

5x 5

5x

x

g f x gxx

( ( ))

3 5

53 5

3xx

5

3 53

xx

xx

5

3 5 3x

x x 5

5x

x

An exponential function is a function of the form

f x ax( )

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

Exponential Functions

3 2 1 0 1 2 3

2

4

6

(0, 1)

(1, 3)

(1, 6)

(-1, 1/3) (-1, 1/6)

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

3 2 1 0 1 2 3

2

4

6

(-1, 3)

(-1, 6)

(0, 1) (1, 1/3) (1, 1/6)

yx

13

yx

16

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

Horizontal Asymptote: y = 2

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

Domain: All real numbers

)

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 rth 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 . I n c a lc u lu s , th i s i se 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

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 l o g i f a n d o n l y i f

If then y x xy log ,3 3

If then b b3 8 3 8 , log

Logarithm Functions

I f t h e n a a u vu v ,

Evaluate: log4 64

y log4 64

4 64y

4 43y y 3

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

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

Properties of the Logarithm Function

implies

Domain: Range:

y x x a

x ya

y

log

0

0

y a x

y xalog

y x

(0, 1)

(1, 0)

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

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

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

Theorem: 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

2 1 1log log loga a ax x x

314

2 1 1log log loga a ax x x

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

Theorem: Change-of-Base Formula

If , and are positive

real numbers, then

a b M 1 1,

loglog

Ma

loglogloga

b

b

MMa

lnln

Ma

Calculate log5 63

logloglog5 63

635

lnln

635

2 574.

A certain bacteria initially increasesaccording to the law of uninhibited growth.A biologist collects the following data:

Day Weight (in grams)0 100.01 123.42 151.93 188.74 230.15 288.16 353.87 433.3

Applications: Exponential Function in Biology

After drawing a scatter diagram of the data, one finds the exponential function of best fit to the data to be:

y x100 02 1234. .

Typically, one then expresses this in base e form as follows.

Express the curve in the form N t N ekt( ) 0

y x100 02 1234. .

N t N ekt( ) 0

N0 100 02 . ekt x1234.

ek 1234.k ln . .1234 0 2103

N t e t( ) . .100 02 0 2103

Predict the population after 9 days.

N t e t( ) . .100 02 0 2103

N e( ) . . ( )9 100 02 0 2103 9 663 9. grams

Year Carbon Emissions1984 51091985 52821986 54641987 55841988 58011989 59121990 59411991 6026

The following data represent the amount of carbon emissions in millions of metric tons:

Applications: Logarithim Function in Pollution Study

y x 2020583 266796 7. ln

Then one can predict the carbon emissions in the year 2000 as:

y 2020583 266796 7 2000 7313. ln

After drawing a scatter diagram of the data, one finds the exponential function of best fit to the data to be:

Solve: log4 3 2 2x

log4 3 2 2x

3 2 42x

3 2 16x 3 18x x 6

Applications: Solution of Logarithmic & Exponential Equations

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: 4 642x

4 642x

4 42 3x x 2 3

x 5

Solve: 9 3 10 02x x

9 3 10 02x x

3 3 3 10 02 2x x

3 9 3 10 02 x 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: 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.

Solve: x ex2 3 This is an example of what is called a transcendental equation. These equations can only be solved graphically or numerically. Estimate solution: rearrange then plot each side of

.3 2 xex

Can you see the solution is about x=-1.78?

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

Applications: Solution of Interest Problems

Theorem: 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

A Prn

nt

1

(a) Annually

A Prn

nt

1

$500

. ( )

10 04

1

1 3

$562.43

(b) Monthly

A Prn

nt

1

$500

. ( )

10 0412

12 3

$563.64

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

Graph

Yx

1

12

100 11212

.

What is the value of y for x = 10?

What is the value of y for x = 20?

Describe the behavior of the graph.

Theorem: Continuous Compounding

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 annum. 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

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

P A

rn

nt

1

If the interest is compounded continuously, then

P Aert

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

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 annum compounded quarterly?

N t Ne kt( ) 0

k > 0

Uninhibited Growth of Cells

More Applications: Solution of Logarithmic & Exponential Equations in Models

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 thelaw of uninhibited growth. If 100 grams bacteriaare present initially, and there are 175 grams ofbacteria after 1 hour, how many will be presentafter 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 Radioactive Decay

A t A e kkt( ) 0 0

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 of is the time required for half of 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

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,

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

The intensity of a sound wave is the amount of energy the wave transmits through a given area.The least intense sound that a human ear can detect at a frequency of 100 Hertz is about 10-12 watt per square meter.

The loudness L(x), measured in decibels of a sound of intensity x is defined as

L x x( ) log 10 10 12

What is the loudness of a rock concert if its intensity is 0.2 watts per square meter?

L xx

( ) log 1010 12

L( . ) log.

0 2 100 2

10 12 113 db

Close to the threshold of pain due to sound which is about 120 db.

The Richter scale converts seismographic readings into numbers that provide an provide an easy reference for measuring the magnitude M of an earthquake.

An earthquake whose seismographic reading measures x millimeters has magnitude M(x) given by

All earthquakes are compared to a zero-level earthquake whose seismographic reading measures 0.001 millimeter at a distance of 100 kilometers from the epicenter.

Mxxx

( ) log100

where x0 = 10-3 is the reading of a zero-level earthquake the same distance from its epicenter.

Determine the magnitude of an earthquake whose seismographic reading is 3 millimeters at a distance 100 kilometers from the epicenter.

M xxx

( ) log100

M ( ) log3 103

10 3 = 3.5

The earthquake measures 3.5 on the Richter scale.