1

Apply Properties of Logarithms

4.5 Apply Properties of Logarithms

• Product Property

• Quotient Property

• Power Property

• Inverse Property

• Change of Base Formula

Apply Properties of Logarithms

Product Rule

Apply Properties of Logarithms

Any base example

Common Log example

Natural Log example

Apply Properties of Logarithms

Quotient Rule

Apply Properties of Logarithms

Any base example

Common Log example

Natural Log example

Apply Properties of Logarithms

Power Rule

2

Apply Properties of Logarithms

Any base example

Common Log example

Natural Log example

Apply Properties of Logarithms

Other Rules Part 1

Apply Properties of Logarithms

Any base example

Common Log example

Natural Log example

Apply Properties of Logarithms

Other Rules Part 2

Apply Properties of Logarithms

Any base example

Common Log example

Natural Log example

Apply Properties of Logarithms

Other Rules Part 3

3

Apply Properties of Logarithms

Any base example

Common Log example

Natural Log example

Apply Properties of LogarithmsExpanding Expressions Using 

Logarithmic RulesExample #1

Apply Properties of LogarithmsExpanding Expressions Using 

Logarithmic RulesExample #2

Apply Properties of LogarithmsExpanding Expressions Using 

Logarithmic RulesExample #3

Apply Properties of LogarithmsExpanding Expressions Using 

Logarithmic RulesExample #4

Apply Properties of Logarithms

Condensing Expressions Using Logarithmic Rules

Example #1

4

Apply Properties of Logarithms

Change of BaseFormula

Apply Properties of Logarithms

Apply Properties of Logarithms

4.5 Apply Properties of Logarithms

• Product Property

• Quotient Property

• Power Property

• Inverse Property

• Change of Base Formula

1

A2.A.19: Properties of Logarithms 1: Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms

1 The expression log12 is equivalent to1) log6 log62) log3 2log23) log3 2log24) log3 log4

2 The expression log4x is equivalent to

1) logx 4

2) 4logx3) log4 logx4) (log4)(logx)

3 Which expression is not equivalent to logb 36?

1) 6logb 2

2) logb 9 logb 4

3) 2logb 6

4) logb 72 logb 2

4 If A r2 , logA equals1) 2log logr2) log 2logr3) 2log 2logr4) 2 logr

5 If L x 2

k, then logL is equal to

1) 2logxk

2) 2(logx logk)3) 2logx logk

4)2logxlogk

6 The expression logb 3

a is equivalent to

1) 3(logb loga)2) log3b loga3) 3logb loga

4)3logbloga

7 If u x

y 2 , which expression is equivalent to logu?

1) logx 2logy2) 2(logx logy)3) 2(logx logy)4) logx 2logy

8 If logx 2 log2a log3a , then logx expressed in terms of loga is equivalent to

1)12

log5a

2)12

log6 loga

3) log6 loga4) log6 2loga

9 The expression log xy is equivalent to

1) 2logx logy2) 2(logx logy)

3)12

logx logy

4)12

(logx logy)

Properties of Logarithms 1 Name: ________________________

Regents Exam Questions A2.A.19: Properties of Logarithms 1 Name: ________________________ www.jmap.org

2

10 If x (82 )( 5), which expression is equivalent to logx?1) 2log8 2log5

2) 2(log8 12

log5)

3) 2log8 12

log5

4) (2log8)(13

log5)

11 If x a b

c, then logx is equal to

1) loga 12

logb logc

2) loga 2logb logc

3) loga 12

logb logc

4) loga 2logb logc

12 Logxy

z is equal to

1)12

logx 12

logy logz

2)12

logx logy logz

3)12

logx logy logzÊËÁÁ

ˆ¯̃̃

4)

12

logxy

logz

13 The expression logxy

w is equivalent to

1)2logxylogw

2) logx logy logw

3)12

(logx logy) logw

4)12

(logxy logw)

14 Logab

is equivalent to

1)12

loga logb

2)12

(loga logb)

3)12

(loga logb)

4)12

loga logb

15 The expression logxn

y

Ê

Ë

ÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃

is equivalent to

1) n logx 12

logy

2) n logx 2logy

3) log(nx) log12

yÊ

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃

4) log(nx) log(2y)

Regents Exam Questions A2.A.19: Properties of Logarithms 1 Name: ________________________ www.jmap.org

3

16 The expression logx 2 y 3

z

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃̃ is equivalent to

1)(2x)(3y)

12

z

2) 2logx 3logy 12

logz

3) log2x log3y log12

z

4) 2logx 3logy 12

logz

17 The expression logx2 y 3

z is equivalent to

1)12

(2logx 3logy logz)

2)12

(2logx 3logy) logz

3) 2logx 3logy logz

4)x 2 y3

z

18 The expression loga3

b is equivalent to

1)13

loga logb

2)13

log(a b)

3) 3loga logb4) 3log(a b)

19 If r A2 BC

3 , then logr can be represented by

1)16

logA 13

logB logC

2) 3(logA2 logB logC)

3)13

log(A2 B) C

4)23

logA 13

logB 13

logC

20 The equation N x 2 y4

z is equivalent to

1) logN 14

(2logx logy logz)

2) logN 14

(2logx logy) logz

3) logN 14

log2x 14

logy logz

4) logN 24

logx 14

log(y z)

21 The expression loga 2

b4 is equivalent to

1)14

loga 2

logb

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜̃̃

2) 4(loga 2 logb)

3)12

(4loga logb)

4)14

(2loga logb)

22 LogcotA is equivalent to1) logsinA logcos A2) logsinA logcos A3) logcos A logsinA4) logcos A logsinA