Warm-upWarm-up 1. Convert the following log & exponential 1. Convert the following log & exponential

equationsequations Log equationLog equation Exponential Exponential

EquationEquation LogLog2 2 16 = 416 = 4 ?? LogLog3 3 1= 01= 0 ?? ?? 5522 = 25 = 25 2. Solve these log expressions:2. Solve these log expressions: LogLog226464 loglog9999 loglog33(1/9)(1/9)

3. Graph this function: f(x) = log3. Graph this function: f(x) = log33(x – 2)(x – 2)

Warm-upWarm-up 1. Convert the following log & 1. Convert the following log &

exponential equationsexponential equations Log equationLog equation Exponential Exponential

EquationEquation LogLog2 2 16 = 416 = 4

LogLog3 3 1 = 01 = 0

?? 5522 = 25 = 25

1624

130

225log5

Warm-upWarm-up 2. Solve these log expressions:2. Solve these log expressions:

LogLog226464

loglog9999

loglog33(1/9)(1/9)

6

1

2

Property of Exponential Property of Exponential EqualityEquality

xxmm = x = xnn ; if and only if m = n ; if and only if m = n

You will use this property a lot when You will use this property a lot when trying to simplify.trying to simplify.

Example 1: Solve Example 1: Solve

64 = 264 = 23n+13n+1

We want the same base (2). Can we write We want the same base (2). Can we write 64 as 264 as 2??

64 = 2x2x2x2x2x2 = 264 = 2x2x2x2x2x2 = 266

2266 = 2 = 23n+13n+1

6 = 3n + 16 = 3n + 1 3n = 53n = 5 n = 5/3n = 5/3

Example 2: Solve Example 2: Solve

55n-3n-3 = 1/25 = 1/25 We want the same base (5). Can we write We want the same base (5). Can we write

1/25 as 51/25 as 5??

25 = 5x5 = 525 = 5x5 = 522

1/25 = 1/51/25 = 1/522 = 5 = 5-2-2

55n-3n-3 = 5 = 5-2-2

n – 3 = -2n – 3 = -2 n = 1n = 1

Using Log Properties to Using Log Properties to Solve EquationsSolve Equations

Section 3-3Section 3-3

Pg 239-245Pg 239-245

ObjectivesObjectives

I can solve equations involving log I can solve equations involving log propertiesproperties

3 Main Properties3 Main Properties

Product PropertyProduct Property Quotient PropertyQuotient Property Power PropertyPower Property

Product Property of Product Property of LogarithmsLogarithms

nmmn bbb logloglog

Example Working Example Working BackwardsBackwards

Solve the following for “x”Solve the following for “x” loglog44 2 + log 2 + log44 6 = log 6 = log44 x x loglog44 2 2•6 = log•6 = log44 x x 2•6 = x2•6 = x x = 12x = 12

Product PropertyProduct Property

1)3log(log zz x333 log7log2log

1)3(log zz

10)3( zz

01032 zz

0)2)(5( zz2,5z

x33 log14log

x14

Quotient Property of LogsQuotient Property of Logs

nmn

mbbb logloglog

Working BackwardsWorking Backwards

LogLog33 6 - Log 6 - Log33 12 12 LogLog33 6/12 6/12 LogLog33 1/2 1/2

Condensing an expressionCondensing an expression

Quotient PropertyQuotient Property8log)3(log4log 222 x 2log)12log()3log( xx

8log3

4log 22

x

83

4

x

)3(84 x

2484 x

x2

5

2log12

)3(log

x

x

212

)3(

x

x

1223 xx

243 xx

x3

5

Quotient Property Quotient Property BackwardsBackwards

Solve the following for xSolve the following for x loglog55 42 – log 42 – log55 6 = log 6 = log55 x x loglog55 42/6 = log 42/6 = log55 x x x = 42/6x = 42/6 x = 7x = 7

Power Property of LogsPower Property of Logs

mpm bp

b loglog

Example Power PropertyExample Power Property

4 log4 log55 x = log x = log55 16 16 loglog55 x x44 = log = log55 16 16 xx44 = 16 = 16 xx44 = 2 = 244

x = 2x = 2

Power PropertyPower Property3

5 25log 83 27log

25log3 5

23

6

27log8 3

38

24

PracticePractice

125log3

1log 55 m

3

1

55 125loglog m

5loglog 55 m

1m

PracticePractice

x5log5log5

16log48log 6666

x5log5log

51648

log 666

x5log5log15log 666 x5log515log 66 x5log75log 66

x575

x15

PracticePractice

49log2

116log

4

1log y

2

1

4

1

49log16loglog y

7log2loglog y

14loglog y

14y

PracticePractice16log4loglog3 555 x

16log4loglog 553

5 x

16log4

log 5

3

5 x

16log4

log 5

3

5 x

164

3

x

643 x

4x

PracticePractice

2)21log(log aa2)21(log aa

100212 aa

0100212 aa

0)4)(25( aa

4,25a

PracticePractice

22log)2(log 62

6 b

22)2(log 26 b

3642 2 b

162 b4b

4b

PracticePractice0)1(log)55(log 2

33 zz

01

)55(log

23

z

z

11

)55(2

z

z

155 2 zz

650 2 zz

160 zz

1,6 z

HomeworkHomework

WS 6-3WS 6-3