Objectives:1. Be able to identify the properties of logarithms. 2. Be able to simplify logarithms by using the properties. 3. Be able to expand logarithms by using the properties. 4. Be able to evaluate logarithms by using the properties.

Critical Vocabulary:Identity PropertyProduct PropertyQuotient PropertyPower Property

I. Logarithmic Properties: a. Identity Property

1log MM

Example:

6log6x66 1x

b. Product Property

log log logb b bMN M N Example 1: 8log6 2log4log 66 Example 2: x3log9 x99 log3log

c. Quotient Property

log log logb b b

M

NM N

Example 1:

5log2

x5loglog 22 x

d. Power Property

Example 1:

e. Pointless Property

Example 2:

Example:

log logbK

bM K M7

16log X X16log7

78 5log 7

1

8 )5(log 5log7

18

log logb bNN

1

x

1log6

x6log

I. Logarithmic Properties:

II. Writing Single Logarithms (Condense)

2log7log Example 1:

14log (Product Property)

Example 2: zyx 222 logloglog4 zyx 22

42 logloglog (Power Property)

zy

x2

4

2 loglog (Quotient Property)

y

zx4

2log (Product Property)

yx 55 loglog4

1Example 3:

yx 54

1

5 loglog (Power Property)

y

x 4

1

5log (Quotient Property)

y

x4

5log (Simplify)

II. Writing Single Logarithms (Condense)

35 )3(log x

III. Expand Each Logarithms

Example 1:

3

42

6logc

ba

36

426 log)(log cba

36

46

26 logloglog cba

cba 666 log3log4log2 (Power Property)

(Product Property)

(Quotient Property)

Example 2:

)3(log3 5 x (Power Property)

)log3(log3 55 x

x55 log33log3

(Product Property)

(Distribute)

IV. Evaluating Logarithms

Example 1:

3log236log 22

2log8log 22 4log2

42 x222 x

2x

(Quotient Property)

(Convert to Exponential)

Example 2:

9log36log 22 4log2

42 x222 x

2x

(Power Property)

(Quotient Property)

(Convert to Exponential)

Theorem: If M=N, Then log logb bM N

•This means if you take the log of both sides the problem does not change.

log 2 7Example: 72 x

7log2log x

7log2log x

2log

7logx

(Convert to Exponential)

(Take log of both sides)

(Power Property)

807354922.2x

(Division)

Page 510-511 #3-41 odds

(20 problems)