Unit 5: Logarithmic FunctionsInverse of exponential functions.

xby

lExponentia

:

62 xEx 1:

•Domain:

•Range:

xy

cLogarithmi

blog

:

•Domain:

•Range:

x712 Ex 2:

Logarithmic and Exponential Conversions

xba log ba x Convert each log expression into an exponential expression.

364

1log4

3

13log27 2144log12 1. 2. 3.

Convert each exponential expression into a log expression.

3662 pmn 32

12 5 4. 5. 6.

(1) Base is always the base (2) Exponent and Answer switch

Example 1 CONVERSION PRACTICE

a) __________________1624

b) __________________8134

c) ______________ 264log8

d) ______________ 42401log7

Exponential Logarithmic

f) __________________125

15 3

f) ______________ 41024log4

Example 1: Continued (Fill In The Blanks)

a) __________________64

1

4

13

c) __________________729

19 3

b) ______________5

16log7776

d) ______________

481

1log3

Exponential Logarithmic

f) __________________170

f) ______________

500001.0log10

Useful Log Properties: MEMORIZE THEM!!!Exponential Reasoning

[1] 01log a

[2] 1log aa

[3] xa xa log

[4] Cannot take logs of negative number

[5] “change of base formula” (for calculator)a

bba log

loglog

)(log negativesnoa

OPERATION PROPERTIES OF LOGARITHMS

#1) Product Property:

#2) Quotient Property:

#3) Power Property:

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)

Useful Log Properties: Examples

xx ;0)(log4

14log

9log

[1] yy;1log2

bb ;1)(log7[2] mm;5log5

zz ;2)3(log3[3] xx;)8(log2

[5] 21log16

OPERATION PROPERTIES OF LOGARITHMSEXAMPLES

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

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?”

828log2 xx

3

22

823

x

x

x

38log2

Example 2 Evaluate using properties (algebraic proof)

a) 4log2c)

2

1log2

e) 2log8

b) 27log3

d) 3/25 5log f)

81

16log

3

2

Solve Exponential Equations with Logs•Solve the exponential until form, bx = a.•Clearing Bases Using Log Conversion•Some answers cannot be evaluated by hand and require calculatora) 1581)2(3 x b) 1180100)6(5 x

a)3

4log8 x b)

2

5log4 x

c) 3log6 x d) 3log2 x

Solving LOG Equations and Inequalities**SIMPLIFY all LOG Expressions**

CASE #1: LOG on one side and VALUE on other Side•Apply Exponential Conversion•Solve (For inequalities x < # requires 0 < x < # because of domain

a) )64(log)3(log 22 xx

c) 273log27log 66 xx

Solving LOG Equations and Inequalities**Simplify all LOG Expressions**

CASE #2: LOG on BOTH •Bases of both sides should be the same•Set the insides of logs equal and Solve

)4(log)3(log 52

5 xx b)

Practice: Solving Logs29log6 x )3(16log4 x

xx aa 2log)8(log 2

3.

1.

5.

2.

4. )136(log)7(log 22 xx3)13(log5 x

Log Property Practice• Condense each Log Expression

xaa log5log yx aa loglog3

cba 555 log3log4log2

1. 2.

3.

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

APPLYING LOG PROPERTIES: SOLVING with PRODUCT PROPERTY

20log10loglog 555 x[a] 2)6(loglog 44 xx[b]

2)5(loglog 66 xx[d]

16log4log)2(log 222 x[c]

10log8loglog 666 x[a]

APPLYING LOG PROPERTIES: SOLVING with QUOTIENT PROPERTY

2)6(loglog 33 xx[b]

APPLYING LOG PROPERTIES: SOLVING with POWER PROPERTY

16log4loglog3 555 x[b]9log)1(log2 33 x[a]

39loglog4 33 x[b]

GENERAL PRACTICE7log3log)52(log 333 x[a]

4)14(log3 x[c] )83(log2log)34(log 555 xx[d]

GENERAL PRACTICE: Continued

25log)3(log2 44 x[e]

6log3log3log)52(log 2222 xx[f]