6.2 Logarithmic Functions

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Transcendental

Functions6.2 Logarithmic Functions

RA Idoy

MATH17



• Logarithmic Function with base

b

• Graph of Logarithmic Function

• Natural Logarithmic Function

• Properties of Logarithmic

Functions

• Exponential and Logarithmic

Equations



Logarithmic Function with

base b

Definition:

The logarithmic function with

base b is the inverse of the

exponential function with base

b.

Note:

log if and only if

y

b y x x b

Dom f Rng f




base b

Examples:

2

33 9 log 9 2

1/2

1/16

1 1 1 1log

16 4 4 2

32 8

2 15

25




base b

More examples:

Find the values of the ff:

Solution:

Therefore,

7log 497

2

log 49

7 497 7

y

y

y

7log 49 2




base b

More examples:


Solution:

Therefore,

5log 55

1/2

log 5

5 5

5 5

y

y

y

5log 5 1/ 2




base b

More examples:


Solution:

Therefore,

6

1

log 66

1

log 1/ 6

6 1/ 6

6 6

y

y

y

6log 1/ 6 1




base b

More examples:


Solution:

Therefore,

3log 813

4

log 81

3 81

3 3

y

y

y

3log 81 4




base b

More examples:


Solution:

Therefore,

10log 0.001

10

3

log 0.001

10 0.00110 10

y

y

y

10log 0.001 3




base b

Other examples:

Solve the given equation for either

x or b.

Solution:

6log 2 x

6

2

log 2

6

36

x

x

x




base b

Other examples:


x or b.

Solution:

27log 2 / 3 x

27

2/3

23

log 2 / 3

27

27

9

x

x

x

x




base b

Other examples:


x or b.

Solution:

log 4 1/ 3b

1/3

31/3 3

log 4 1 / 3

4

4

64

b

b

b

b




base b

Other examples:


x or b.

Solution:

log 81 2b

2

1/22 1/2

log 81 2

81

81

1/ 9

b

b

b

b




base b

Recall:

Since the logarithmic function with

base b is the inverse of exponential

function with base b, then

1

1

f f x x

f f x x

log

log

b

xb

x

b x

b x




base b

Example:

5

2log 2 5

3

10log 10 3



Graph of Logarithmic

function of base b

Since logarithmic function is

the inverse of exponential

function, then its graph is the

reflection of the latter

function at y=x.



x y b

0b

logb

y x



x y b

0 1b

logb

y x



Natural Logarithmic

Function

Definition:

The natural logarithmic

function is the inverse of the

natural exponential function.

ln if and only if y y x x e

Dom f Rng f



Natural Logarithmic

Function

Note:

ln 1e ln x

e x

lnx

e x



Natural Logarithmic

Function

Example 1: Recall the function

obtained from exponential

growth of bacteria culture.

Determine how many minutes

elapse until there are 30,000bacteria present in the

culture.

0.041500

t f t e



Natural Logarithmic

Function

Solution:

Let T be the

time elapse

until there are30,000 bacteria

present

Given:

0.041500 T f T e

30,000 f T 0.04

0.04

0.04

30000 150020

ln 20 ln

ln 20 0.04

74.9

T

T

T

ee

e

T

T



Natural Logarithmic

Function

Conclusion:

Therefore 74.9 minutes or 1

hr, 14 min and 54 sec elapse

until there are 30,000 bacteria

present.



Natural Logarithmic

Function

Example 2:

If $1000 is deposited into a

savings account that pays an

annual interest rate of 6percent compounded

continuously, and no

withdrawals or additional

deposits are made, how long

will it take until there is

$1500 on deposit?



Natural Logarithmic

Function

Solution:

Let T years

be the time to

take toaccumulate $1500

Recall: it A Pe

0.061500 1000

T e

0.06

0.06

0.06

1500 1000

1.5ln1.5 ln

ln1.5 0.066.758

T

T

T

e

e

e

T T



Natural Logarithmic

Function

Conclusion:

Therefore it takes 6.758

years or 6 years, 9 months and

3 days for the initial deposit

to accumulate to $1500.



Properties of Logarithmic

Functions

Theorem:

If b>0, b≠1, and u and v are

positive numbers, then

log log logb b b

uv u v

log log logb b b

u u vv




Functions

Theorem:

If b>0, b≠1, and u and v are

positive numbers, then

log logn

b bu n u




Functions

Examples: Express each of the

following in terms of log of x, y

and z, each of which represents a

positive number.2 3 4

logb

x y z

2log

b

x

yz

2

5

3log

b

xy

z




Functions

Examples: Write each of the

following expression as a single

logarithm with a coefficient of 1.

log 2log 3logb b b

x y z

1 log 4 log 3 log log3

b b b b x y



Exponential and

Logarithmic Equations

Examples: Find the solution set.

1.

2.

3.

4.5.

3 16

x 3 1

5 0.08 x

17 3

x x

4log 19 10

log 3 2 x



Exponential and


Examples: Find the solution set.

6.

7.

8.

2 2

log 4 log 3 3 x x

3 3log log 2 3 3 x x

3 3 4 x x



Exponential and


Suppose on January 1, 1988, the

population of a certain city was

800,000. From then until the year

2000 the population is expected

to increase at the rate of 3.5

percent per year. Therefore t

years after January 1, 1988, the

population is expected to be800,000(1.035)t, where 0≤t≤12.When would you predict the

population will be one million?



Exponential and


• We wish to

determine the

value of t

800000 1.035 1000000

1.035 1.25

t

t

log 1.035 log1.25

log 1.035 log1.25log1.25

log1.035

6.5

t

t

t

t



Exponential and


Conclusion:

Therefore, after six and one-

half year, the population will

be one million. This is on July

1, 1994.



Exponential and


If $1000 is deposited in a

savings account that pays an

annual interest rate of 6

percent compounded quarterlyand no withdrawals or

additional deposits are made,

how long will it take untilthere is $1500 on deposit?



Exponential and


0.061500 1000 1

4

n

1.5 1.015

log1.5 log 1.015

log1.5 log 1.015

log1.5

log1.01527.23

n

n

n

n

n

1

n

n

i A P

m



Exponential and


Conclusion:

Therefore, it will take 28

quarters until there is $1500

on deposit.



Assignment (1/2)

Find the solution set of the

following equations (show

solutions)

1.

2.

3.

4.

5.

2 1 3 13 5 x x 1.04 0.932

x

8log 28 2 2log 11 log 1 3 x x 4 4 3

x x

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6.2 Logarithmic Functions

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