Unit 5: Properties of Logarithms MEMORIZE THEM!!!

Exponential Reasoning[1] 01log a

[2] 1log aa

[3] xa xa log

[4] Cannot take logs of negative number

10 a

aa 1

xx aa

)(log negativesnoa

[3b]xa xa log

Useful Log Properties: Examples

xx ;0)(log 4

14log

9log

[1] yy;1log 2

bb ;1)(log 7[2] mm;5log 5

zz ;2)3(log 3[3] xx;)8(log 2

[5] 21log16

Base – Base (Inverse property: True for all logs that have same base of log as base of power)

a)

21lne12ln xe

b) c)9log44 x43

3log )25(log88 x

d) e) f) 976

6log2 x7)5(log

3

4 65 311 )6(log11 x

g) h) 43ln xei)

OPERATION PROPERTIES OF LOGARITHMS

#1) Product Property: )(logloglog mnnm bbb

#2) Quotient Property:n

mnm bbb logloglog

#3) Power Property: pbb mmp loglog

Log of a product is equal to the SUM of the logs of both multipliers of the same base

Log of a quotient “fraction” is equal to the DIFFERENCE of the logs of the numerator and denominator

Log of a power statement is equal to the MULTIPLICATION of the power (p) times the log of the power’s base (m)

OPERATION PROPERTIES OF LOGARITHMSCondense: Expand

5log4log 33

6log27log 22

(1a)

5

2

13 6log

x8log7

(2a) 9

14log8

(1b)

(3a) 11log2 9

(2b)

(3b)

Expand Each Logarithm Using Properties

36 4log53log2

6

11log7

(1)

(7)

(3)

yx3log5(5)

73 2log x

(2)

r

p7log(6)ab4log(4)

(8) 45 3log x

5log

2

6

x(9)

Condense Each Logarithm Using Properties

6log8log 22 5

30log7(1)

(4)

(5)

(2)

)log2(log2 yx(6)

(3) 3loglog4 22 x 7log2log5log 333

3

7log2 4

Log Property Practice• Condense each Log Expression

xaa log5log yx aa loglog3

cba 555 log3log4log2

1. 2.

3.

NATURAL LOG PROPERTIES: All Log Properties work for Natural Logs because its just a special notation for base e

a) )5ln(2)ln(3 x b)

c) d)

)23ln()2ln(4 2 xx

)3ln()11ln( xx )ln(7)2ln()ln(5 yzx

Evaluating Log Expressions: General Rules

1) Set the log expression equal to x

2) Convert log to exponential form

3) Solve the resulting exponential equation for x.

x8log2

8log2 “2 raised to what power equals 8?”

828log 2 xx

3

22

823

x

x

x

38log2

Example 2 Evaluate using properties (algebraic proof)

a) 4log2c)

2

1log2

e) 2log8

b) 27log 3

d) 3/25 5log f)

81

16log

3

2

Use the given values and log properties to evaluate45.2log a

6loga 15loga

3

2loga

68.3log a30.15log a

4. 6.5.

7. 8.

20loga

9

10loga 8. 125loga