lat03 09

(1)

(2)

Chapter 9

(3)

9.1

(4)

Additional Properties of Exponents



If any real number

a

> 0,

a



1, the following

statements are true.



a

x

is a unique real number for all real numbers

x

.



a

b

= a

c

if and only if

b

=

c

.



If

a

> 1 and

m

<

n

, then

a

m

< a

n

.

(5)

Evaluating an Exponential Expression



If

f

(

x

) = 3

x

,

find each of the following.



a)

f

(



1)

b)

f

(3)

c)

f

(3/2)

d)

f

(5.01)



Solutions

:



a)

b)



c)

d)

1

1 ( ) 3

3

1 f









f

( ) 3

3 

27

3

1

3

2

3

3 (3 )

2 f

   

_{ }

 

5.01 (5.01) 3

245.68

(6)

(7)

(8)

Characteristics of the Graph of

f

(

x

) =

a

x



1. The points (0, 1), and (1,

a

) are on the

graph.



2. If

a

> 1, then

f

is an increasing function; if 0 <

a

<

1,

then

f

is a decreasing function.



3. The

x

-axis is a horizontal asymptote.



4. The domain is (



,



), and the range is (0,



).

1 1,

,

a



_







(9)

Example: Graphing



Graph

f

(

x

) = 6

x

.



y

-intercept = 1



x

-axis = horizontal

asymptote



Domain: (



,



)



Range (0,



)

1.09

0.5

1

0

0.16 

1 f

(

x

)

(10)

Graphing Reflections and Translations



Graph each function.



a)

f

(

x

) =



3 x



b)

f

(

x

) = 3

x +

2

(11)

Solution



f

(

x

) =



3 x



Reflected across the

x

-axis.

(12)

Solution



f

(

x

) = 3

x +

2 

The graph of

f

(

x

) = 3

x

(13)

Solution



f

(

x

) = 3

x

+

2 

The graph of

f

(

x

) = 3

x

(14)

Exponential Equations



Solve

1 ₈₁

4 .

  

 

 

x

1

3

1

81

4 (4 )

81

4

81

3

3 Def. of negative exponent

(

)

4

4 Write 81 as a power of 4

Property (b)

Multiply b 1

y

(15)

Another Example



Solve

3 x

+ 1

= 27

x



3

1

3

1

3

1

3

9

3

27

3 Write 27 as a power of 3.

(

3

3 ( )

3

3 1 3

9

2

10

5 )

Property (b)

Subtract 3 and 1.

Divide by 2.

x

m

n

mn

(16)

Compound Interest



If

P

dollars are deposited in an account paying

an annual rate of interest

r

compounded (paid)

m

times per year, then after

t

years the account

will contain

A

dollars, where

1 .

tm

r

A P

m







_



_

(17)

Example



Suppose $2000 is deposited in an account

paying 6% compounded semiannually (twice a

year).



a) Find the amount in the account after 10 years.



b) How much interest is earned over the 10-yr

(18)

Solution

a)



Thus, $3612.22 is an account after 10 yr.

b) The interest earned for that period is

$3612.22



$2000 = $1612.22

(19)

Definitions



Value of

e

:

To nine decimal places,

e



2.718281828



Continuous Compounding

If

P

dollars are deposited at a rate of

interest

r

(20)

Example



Suppose $2000 is deposited in an account

paying 2% interest compounded continuously for

3 yr. Find the total amount on deposit at the end

of the 3 yr.



Solution

:

.02(3)

.06

2000

2000(1.061837)

2123.67

rt

A Pe

e

(21)

9.2

(22)

Logarithm



For all real numbers

y

, and all positive numbers

a

and

x

, where

a



1:



Meaning of log

_a

x

A logarithm is an exponent; log

_a

x

is the exponent

to which the base

a

must be raised to obtain

x

.

log

_a

if and only if

y

.

(23)

Solving Logarithmic Equations



Solve each equation.



a)

b)

3

27 log

3

64

27

64

3

4

3

4 x

x



_{ }











 

  

_{ }



9 3 / 2

1/ 2 3

(24)

Logarithmic Function



If

a

> 0,

a



1, and

x

> 0, then

defines the

logarithmic function

with base

a

.

Logarithms can be found for

positive numbers

only.

( ) log

_a

(25)

(26)

(27)

Characteristics of the Graph of

f

(

x

) = log

_a

x



The points (1, 0), and (

a

, 1) are on the

graph.



If

a

> 1, then

f

is an increasing function; if

0 <

a

< 1, then

f

is a decreasing function.



The

y

-axis is a vertical asymptote.



The domain is (0,



), and the range is (



,



).

1 , 1 ,

a



_







(28)

Example: Graphing



Graph



Write

in exponential form

as

Now find some ordered pairs.

1/ 4

( ) log



f x

x

1/ 4

( )

log

f x

 

y

x

1

4 y

x

  

 

 

2 1/16

0

1

(29)



Graph



Write

in exponential form

as

Now find some ordered

pairs.

Example: Graphing

5 ( ) log



f x

x

1

5

0

1 y

x

5 y

x



5 ( ) log

(30)

Translated Logarithmic Functions



Graph the function.



The vertical asymptote is

x

= 1.



To find some ordered pairs,

use the equivalent exponent

form.



3 ( ) log (





1)

f x

x

3 log (

1)

1 3

3

1 y

y

x





 

(31)

Translated Logarithmic Functions

continued



Graph



To find some ordered

pairs, use the equivalent

exponent form.



4 ( ) (log ) 1





f x

x



4 

4

1 log

1 1 log

4 y

y

x

y

x







(32)

Properties of Logarithms, For

x

> 0,

y

> 0,

a

> 0,

a



1, and any real number

r

:

The logarithm of a number raised to a

power is equal to the exponent multiplied

by the logarithm of the number.

Power Property

The logarithm of the quotient of two

numbers is equal to the difference

between the logarithms of the numbers.

Quotient Property

The logarithm of a product of two

numbers is equal to the sum of the

logarithms of the numbers

Product Property

Description

Property

log

_a

xy



log

_a

x



log

_a

y

log

_a

x

log

_a

x

log

_a

y





(33)

Using the Properties of Logarithms



Rewrite each expression. Assume all variables represent

positive real numbers with

a



1 and

b



1. 

a)



b)



c)

6

12 log

7

4 log

11

2 log

_a

abc

w

6

6 log

12 log

12

7

7 



log

4

4 1/ 2

log

11 log (11)

1 lo

2 g 11



2

2 log

_a

abc

log

_a

a

log

_a

b

log

_a

c

log

_a

w

(34)

Using the Properties of Logarithms



Write each expression as a single logarithm with coefficient 1.

Assume all variables represent positive real numbers with

a



1 and

b



1 .



a)



b)

3

3 log (

x

 

1) log

x



log 5

3log

_y

r



5log

_y

t

3

3 log

1 log

log

5 log

(

1)

5 (

x



)



x





x



x

3

5

5 log

log

l

5

3 og

y

(35)

Theorem on Inverses



For

a

> 0,

a



1:



By the results of this theorem:

log

_a

x

_{and log}

x

_.

a



x

a



x

7

1 log

0

(36)

9.3

Evaluating Logarithms;

(37)

Common Logarithm



For all positive numbers

x

, log

x

= log

₁₀

x

.



A calculator with a log key can be used to find

the base 10 logarithm of any positive number.



In chemistry, the

pH

of a solution is defined as

pH =



log[H

₃

O

+

],

where [H

3 O

+

] is the hydronium

(38)

Example: Finding pH



Find the pH of a solution with [H

₃

O

+

] = 3.1



10 

4 

pH =



log[

H

₃

O

+

]

=



log(

3.1 

10 4

₎

=



log(3.1 + log 10



4 )

=



(.4914



4)

=



.4914 + 4

(39)

Loudness of Sound



The loudness of sounds is measured in a unit

called a

decibel

. To measure with this unit, we

first assign an intensity of

I

₀

to a very faint

sound, called the

threshold sound

. If a particular

sound has intensity

I

, then the decibel rating of

this louder sound is

0 10log

I

.

d

I

(40)

Example: Decibel



Find the decibel rating of a sound with intensity

1000

I

₀

.

The sound has a decibel rating of 30.

0

0 1000

10log

.

10log

10log1000

10(3)

30 I

d

I

d

I



(41)

Natural Logarithm



For all positive numbers

x

, ln

x

= log

_e

x



Natural logarithms can be found using a

(42)

Example: Age of Rocks



Geologists sometimes measure the age of rocks

by using “atomic clocks.” by measuring the

amounts of potassium 40 and argon 40 in a rock,

the age

t

of the specimen in years is found with

the formula



Where

A

and

K

are the numbers of atoms of

argon 40 and potassium 40, respectively in the

specimen.



9 

ln(1 8.33

1.26 10

,

ln 2

A

K

t

 



_{ }

 

(43)

Example continued



How old is a rock in which

A

= 0 and

K

> 0?



If

A

= 0 and

A/K

= 0



The rock is new (0 yr old).



The ratio

A

/

K

for a

sample of granite from

New Hampshire is .212.

How old is the sample?



Since

A/K

= .212, we

have



The granite is about 1.85

billion yr old.



9 

9 1.26 10

ln 2

(1.26 10 )( ) 0

ln1

0 t















9

9 ln[1 8.33(

)]

1.26 10

ln 2

1 .21

.85 1

2

0 t







(44)

Change-of-Base Theorem



For any positive real numbers

x

,

a

, and

b

, where

a



1 and

b



1:

log

.

log

b

a

b

x

a

(45)

Examples

a) log

₅

12 b) log

₂

.4

Use the change-of-base theorem to find an

approximation to four decimal places for each logarithm.

(46)

Properties of Logarithms



If

x

> 0,

y

> 0,

a

> 0, and

a



1, then

(47)

Example: Solving an Exponential

Equation



Solve

8 x

= 15



The solution set is {1.3023}.

8

15

8

15 ln8 ln15

ln15

ln

8 1.3023

x



(48)

Example: Solving an Exponential

Equation



Solve

5

2 x



1 

.3

x



3 continued

2

1

3

2

1

3

5 .3

(2

1)ln 5 (

3)ln.3

2 ln 5 ln 5

ln.3 3ln.3

2 ln 5

ln.3 3ln.3 ln 5

(2ln 5 ln.3) 3l

ln

n.3 ln 5

x



































3ln.3 ln 5

2ln 5 ln.3

ln.027 ln 5

(49)

Example: Solving Base

e



Solve

e

x

3 

300

3

300 ln

ln 300

300 1.7867



 



x

(50)

Logarithmic Equations



Solve

log (



2) log (



 

1) log (



2)

b

x

b

x

b

x





2

2 log (

2) log (

1)(

2)

(

2)

3

2

0

4

0 (

4)

0 4 0

0

4 b

x

b

x

x x

x

or

x

























 

(51)

Example: Solving a Logarithmic

Equation



Solve

2

5 log

log

l

8

4

5

8

4 8(

4) 5

8 32 5

9 og (

2

5 ) log

(

4

3

(52)

Example: Solving a Logarithmic

Equation



Solve

continued

2

4 ln

ln(8

)

1

4

8

1 (8

)(

1)

4

8 ln(

4) ln(

1) ln

)

4

9 (

8

4

8

8 



_{ }

_



_









_{ }



  

 



 



 







  

x

x x

x

x x

x

0

2

8

12 0 (

6)(

2)

6 0

2 0

6

2 x

x

or

x

or

x











 



(53)