Name : ______ ( ) Class : Date :_

Name : ______________ ( ) Class : ________ Date :_________

Objectives:

Unit 7: Logarithmic and Exponential Functions

Graphs

Solving Equations of the Form

Logarithmic Equations

Laws of Logarithms

Logarithms

Common and Natural

Logarithms

xa b

Graphs of Exponential Functions

Graph ofxy a

x

y

-3

-2

-1

1

2

3

4

5

6

7

Graph ofxy e

Graphs of Exponential Functions

x

y

-6

-5

-4

-3

-2

-1

1

2

3

4

Graph of

lny x

x

y

-6

-5

-4

-3

-2

-1

1

2

3

4

Graph of

y = lg x

Graphs of Common Logarithms

Convert to logarithmic form.

3

1log 2

9

2 13

9 The logarithm or

index for the given base is -2.

Convert to index form.

4log 64 3

34 64

The base is 3.

The base is 4. The logarithm or index for the given

base is 3.

Logarithms

If a logarithm is defined for base a, then

log 1a a

and

log 1 0a

1since a a

0since 1a

Special Cases

Logarithms

Evaluate the following.

2 5log 1 4log 5

2

5

3log 2

4 2log 1x x

0 4 1

4

23

2

1

0

2

4

25 25

4 16

Logarithms

Example 1:

A common logarithm is a logarithm to the base 10.

10log is commonly abbreviated to lgx x

Tables of common logarithms were often used for calculating in

the days before the electronic calculator.

On a scientific calculator, common logarithms can be evaluated using the LOG key.

Logarithms

A natural logarithm is a logarithm to the base e.

elog is commonly abbreviated to lnx x

Natural logarithms are also known as

Naperian logarithms after

John Napier (1550 - 1617).

On a scientific calculator, natural logarithms can be evaluated using the LN key.

Logarithms

From the definitions of logarithms, the following statements are equivalent.

lg 10xy x y

Let’s use these definitions in some

examples.

and

ln exy x y

Logarithms

Convert the following to index form.

lg1000 3

ln 3 m

3 1000 10

3 em

Convert the following to logarithmic form.

2 10 0.01 lg 0.01 2

2 e x k ln 2k x

Index form

Index form

Logarithmic form

Logarithmic form

Logarithms

Example 2:

Find y in terms of x.

ln 1y x

1 exy

2lg 2y x

2lg

2

xy

21

2 210 10x x

y

e 1xy

Index form

Index form

Rearrange

Rearrange

Alternate form

Surds, Indices and Logarithms

(a)

(b)

Example 3:

Solve for x.

ln 4 lg3 lg5x

ln 4 0.4771 0.6990x

0.33354 ex 4 1.396x

0.349x

ln 4 0.3335x

Evaluate using the calculator.

Index form

In most calculators, the function ex is on

the same key as LN.

Evaluate and solve for x.

Logarithms

Example 4:

Solve for x.

2lg 2 lg3x

2 lg 2 0.4771x

0.22762 10x 2 1.689x

3.69x

lg 2 0.2276x

Evaluate using the calculator.

Index form

In most calculators, the function 10x is

on the same key as LOG.

Evaluate and solve for x.

Logarithms

Example 5:

The Power Law

If and are positive numbers and 1, then

log log for any real number .ra a

a x a

x r x r

If , and are positive numbers and 1, then

log log log .a a a

a x y a

xy x y

The Product Law

If , and are positive numbers and 1, then

log log log .a a a

a x y a

xx y

y

The Quotient Law

Logarithms

Let’s use these laws in some

examples.

If , and are positive numbers and 1, 1, then

loglog .

logc

ac

a b c a c

bb

a

The Change of Base Law

1log .

logab

ba

A special case

Logarithms

Example 6:


3 3 3log 4 log 2 log 72

Combine using the product and quotient laws.

2log 5 3log 2 log 4x x x

3

4 2log

72

23log 3

32log 32

2 3log 5 log 2 log 4x x x 2

3

5 4log

2x

25log

2x

Combine using the product and quotient laws.

Apply the power law.Apply the

power law.

Logarithms

(a)

(b)

4 4Given that log 3 and log 5

express the following in terms of and .

a b

a b

Separate using the product and quotient laws.

4log 45 24og 5l 3

24 4log 3 log 5

Apply the power law.

4 42log 3 log 5

2a b

Logarithms

Example 7:

2 23 log log 2 x y x y

Combine, applying the quotient law.

2 23 log 2 logx y x y

2

23 log

x y

x y

Rearrange and solve the equation.

322

x y

x y

Find y in terms of x.

28

x y

x y

2 8 8x y x y

10 7y x 0.7y x

Arrange the log terms on one

side.

Index form

Logarithms

Example 8:


lg 4 lg10lg5 lg 2

lg 10lg 25

Apply the

change of base law.


5 2

25

log 4 log 10

log 10

lg 4 lg10 lg 25

lg5 lg 2 lg 10

12

2 2lg 2 lg10 lg5

lg5 lg 2 lg10

12

2lg 2 lg10 2lg5

lg5 lg 2 lg10 8

Express as powers of 2, 5

and 10.

Logarithms

Example 9:

log loga aM N M N

For two logarithms of the same base,

Let’s use this property to solve some logarithmic

equations.

Logarithms

An Important Property of Logarithms

Combine using the

product law.

3 3 3log 2 log 2 log 2 1x x x

Example : 10

3 3log 2 2 log 2 1x x x

23 3log 4 log 2 1x x

2 4 2 1x x 2 2 3 0x x

1, 3x x 3 3But, if 1, then log 2 and log 2 1

are undefined as 2 0 and 2 1 0

x x x

x x

Use the property of logarithms.

Remember to check if the results

are acceptable

.

3 1 0x x

Logarithms

So, x = 3.


Example 11:

Solve the following equation.

Index form

Remember to check if the results are acceptable.

4 2 9log 6 log 8 log 3x

123


4log 6 3 0.5x

4log 6 3.5x 3.56 4x

122x 6 128x

Logarithms

log4(6 – x) is defined for x = –122.

Apply the change of base law.

Example 12 :


Substitute

Both results are

acceptable.

3log 2 3log 3xx

3

13 loglog 2 3 xx

3Let log x u 3then 2 uu 2 2 3u u

2 2 3 0u u 3 1 0u u

3 and 1u u 3 13 and 3x x

127 and 3x x

Logarithms

log loga aM N M N

For two logarithms of the same base,

Let’s use this property to solve some logarithmic

equations.

Logarithms

An Important Property of Logarithms

Combine using the

product law.

3 3 3log 2 log 2 log 2 1x x x

Example 13 :

3 3log 2 2 log 2 1x x x

23 3log 4 log 2 1x x

2 4 2 1x x 2 2 3 0x x

1, 3x x 3 3But, if 1, then log 2 and log 2 1

are undefined as 2 0 and 2 1 0

x x x

x x

Use the property of logarithms.

Remember to check if the results

are acceptable

.

3 1 0x x

Logarithms

So, x = 3.


Example 14 :


Index form

Remember to check if the results are acceptable.


123


4log 6 3 0.5x

4log 6 3.5x 3.56 4x

122x 6 128x

Logarithms

log4(6 – x) is defined for x = –122.

Apply the change of base law.

Example 15 :


Substitute

Both results are

acceptable.

3log 2 3log 3xx

3

13 loglog 2 3 xx

3Let log x u 3then 2 uu 2 2 3u u

2 2 3 0u u 3 1 0u u

3 and 1u u 3 13 and 3x x

127 and 3x x

Logarithms

Documents

Name : ______________ ( ) Class : ________ Date :_________

Name : ______ ( ) Class : Date :_