3 243x

5x

3log 243Solve xRe :logb

membera c

ca b53 3x

We can unite bases! Now bases are same!

1 148

xSolve We can unite bases!2( 1) 12 8x

2( 1) 3( 1)2 2x 2 2 32 2x Now bases are same!

2 2 3x 2 2 3x 2 3 2x 2 5x

52

x

Check (Remember: Back to Original) 52

1 148

x

215 148

1 18 8

true

21927

x

xSolve

We can unite bases!2 23 27x x 2 3( 2)3 3x x 2 3 63 3x x Now bases are same!2 3 6x x

2 3 6x x 5 65 5x

65

x

Check in original219

27

x

x

66 55

21927

13.9666 13.9666

8-4 Solving Logarithmic Equations and Inequalities

2 2log 3 log 2 1Solve x x

Attention Inequality log Domain first. 3 0Domain x 3x

2 1 0x 2 12 2x

0.5x

2 2log 3 log 2 1x x

3 2 1x x 2 1 3x x

2x

21 1x

2x

Reverse the direction when dividing by “minus”:{ 0.5 2}solution x

3x 0.5x From domain before

Check 1 (Remember: Back to Original) 2 2log 3 log 2 1x x

2 2log 3 lo1 (g )2 11

2 2log 4 log 3

2 1.5850 true

2 2log 2 log 6 3Solve x x

Attention Inequality log Domain first. 2 0Domain x 2x

6 3 0x 3 63 3x

2x

2 2log 2 log 6 3x x 2 6 3x x 3 6 2x x 4 44 4x

1x 2x 2x

From domain:

:{1 2}solution x Check 1.5 (Remember: Back to Original) 2 2log 2 log 6 3x x 2 2log 2 log 61.5 (1.5)3

2 2log 3.5 log 1.51.8074 0.5850 true

3 3log 3 4 log 2Solve x x

Attention Inequality log Domain first.3 4 0Domain x 43

x

2 0x 2x

3 3log 4 3 log 2x x 4 3 2x x 4 2 3x x

3 53 3x

53

x

43

x

2x

From domain::{ 2}solution x

15

log 125Solve x Re :logb

membera c

ca b

11255

x

We can unite bases!

35 5 x Now bases are same!3 x

3x

11255

x

11log 7 1Solve x Attention Inequality log Domain first. 7 0Domain x

7x 11log 7 1x

Re :logb

membera c

ca b17 11x

7 11x 11 7x 4x

7x From domain

{ 4 7}solution x

Check 0 (Remember: Back to Original) 11log 7 1x

11 0log 7 1

11log 7 1

0.8115 1 true

8-5 Properties of Logarithms

3 3log 2 log 2Solve x x

Re :

log log logb b b

membermm nn

3 3log 2 log 2x x

3

2log 2xx

22 3x

x

2 9xx

91

2xx

2 9x x 2 9x x 2 8x

14

x

Do Cross Multiply

: ( )14

Check replace in original

3 3log 21 1o42

4l g

3 3

9 1log log 24 4

2 2 true

2 2log 4 5 logSolve x x

2 2log 4 log 5x x

Re :log log logb b b

memberm n m n

2 2log 4 log 5x x

2log 4 5x x

54 2x x

2 2log 4 log 5x x

2 4 32x x 2 4 32 0x x

Re :logb

membera c

ca b

2 4 32 0x x 4 8 0x x

Use MODE 5 3 a = 1, b= -4, c= -32

4 04

xx

8 08

xx

Check -4 (Remember: Back to Original) 2 2log 4 5 logx x 2 2log 4 5 log4 4

Undefined, so ignore -4

Check 8 2 2log 4 5 logx x 2 2log 4 g8 5 8lo

2 5 3 true 2 2log 4 5 log 8

only solution is 8

2 2log 2 3 2logSolve x x

2 2log 2 3 2logx x 2 2

2log 2 3 logx x 22 3x x

2 2 3 0x x

2 2 3 0x x Use MODE 5 3 a = 1, b= -2, c= -3

3 1 0x x 3 03

xx

1 01

xx

Check 3 (Remember: Back to Original) 2 2log 2 3 2logx x

2 2(3) 3log 2 3 2log

2 2log 9 2log 33.1699 = 3.1699

Check -1 (Remember: Back to Original) 2 2log 2 3 2logx x

2 2log 2 3 2log( 1) ( 1)

Undefined, so ignore -1only solution is 3

2

2 2log 9 log 4 6Solve m

Re :log log logb b b

memberm n m n

2

2 2log 9 log 4 6m 2

2log 4 9 6m

Re :logb

membera c

ca b

2 64 9 2m 24 36 64m

24 100m

24 100m 2 25m 2 25m

5m

Square root both sides

Check -5 (Remember: Back to Original) 2

2 2log 9 log 4 6m

2

2 2log 9 log 45 6

2 2log 16 log 4 6 6 6 true

Check 5 (Remember: Back to Original) 2

2 2log 9 log 4 6m 2 2

2log 9 log 4 65

2 2log 16 log 4 6 6 6 trueThe solutions are 5 and -5

2 2 2 2log 3 log log 4 log 4x x Solve. Check your solution.

2 2 2 2log 3 log log 4 log 4x x

Re :

log log logb b b

membermm nn

2 2 2 2log 3 log log 4 log 4x x

2 2

3 4log log4x x

3 44x x

3( 4) 4x x

3( 4) 4x x

3 12 4x x

12 4 3x x

12x

Check 12 (Remember: Back to Original) 2 2 2 2log 3 log log 4 log 4x x

2 2 2 2log 3 log log 4 log 1 412 2

2 2 2 212log 3 log log 4 log 16 2 2 true

2

4 4 4log 4 log 2 log 1x x Solve. Check your solution.Re :

log log logb b b

membermm nn

2

4 4 4log 4 log 2 log 1x x 2

4 4

4log log 12

xx

2 4 12 1

xx

2 4 2x x

Do Cross Multiply

2 4 2 0x x 2 6 0x x Use MODE 5 3

a = 1, b= -1, c= -6 3 2 0x x

3 03

xx

2 02

xx

Check 3 (Remember: Back to Original) 2

4 4 4log 4 log 2 log 1x x

2

4 4 4log 4 log og3 3 2 l 1

4 4 4log 5 log 5 log 1

0 0 true

Check -2 (Remember: Back to Original) 2

4 4 4log 4 log 2 log 1x x

4 4 4

2log 4 log( 2) ( 2) 2 log 1

4 4 4log 0 log 0 log 1 Undefined, so ignore -2only solution is 3

log 12 ?a . 2log 2 log 3b bA . log 5 2log 2a aB

. log 14 log 2a aC

. log 3 2log 2a aD

log 12 ( )b

log 20 ( )a

log 7 ( )a

log 12 ( )a

2 2 2

1 1log log 16 log 254 2

Solve m

1 14 2

2 2 2log log 16 log 25m Raise the powers

1 14 2

2 2log log 16 25m

2 2log log 10m

10m

4 4 4 4

1log 0.25 3log log 64 5log 23

Solve x

Raise the powers13 53

4 4 4 4log 0.25 log log 64 log 2x 1

3 534 4log 0.25 log 64 2x

3

4 4log 0.25 log 128x 3

0.250.25 1

25280.

x

3 512x 33 3 512x

8x

9log 5 log 29 bEvaluate and b9log 595

log 2bb2

5 512log 3 log 2735Evaluate

12 3

5 5log 3 log 275

2

5 13

3log

2755log 35 3

Raise the powers first!

1 1log 4 log 272 3b bEvaluate b

1132

5 5log 4 log 27b

1132log 4 27bb

log 6bb6

Raise the powers first!

3 3 3 3log 5 log 10 log 4 log 2Show that

3 3 3 3 3log 5 log 10 log 4 5log log 410

3

5log 410

3log 2

8-6 Common Logarithms

Express log9 22 in terms of common logarithms. Then approximate its value to four decimal places.9

log22log 22log9

1.4068

Common logarithm change to base 10

Express log5 14 in terms of common logarithms. Then approximate its value to four decimal places.5

log14log 14log5

0.6099

Common logarithm change to base 10

25 21xSolveRound to four decimal places

We can’t unite bases!So, “log” both sides!2log 5 log21x

2 log 5 log21x

2 log 5 log21x Divide by 2log5 !!2log52 log5 log21

2log5x

0.9458x

34 10xSolveRound to four decimal places

We can’t unite bases!So, “log” both sides!3log 4 log10x

3 log 4 log10x

3 log 4 log10x Divide by 3log4 !!3log43 log4 log10

3log4x

0.5537x

36 5xSolve We can’t unite bases!So, “log” both sides!A. 0.2375

B. 1.1132C. 3.3398D. 43.2563

Do the calculations!

3 3log 2 0.6309 log 12Use toapproximate

3 3log 12 log 2 2 3

3 3 3log 2 log 2 log 3 0.6309 0.6309 1 2.2618

3 3

3log 2 0.6309 log2

Use toapproximate

3 3 3

3log log 3 log 22

1 0.6309 0.3691

5log 11 1.4899Use and5 5log 2 0.4307 log 44to find

5 5log 44 log 2 2 11

5 5 5log 2 log 2 log 11

0.4307 0.4307 1.4899 2.3513

5log 3 0.6826Use and5 5log 2 0.4307 log 54to find

5 5log 54 log 2 3 3 3

5 5 5 5log 2 log 3 log 3 log 3

0.4307 0.6826 0.6826 0.6826

2.4785

4log 3 0.7925Use and

4 4

9log 7 1.4037 log7

to find

4 4 4

9log log 9 log 77

4 4log 3 3 log 7 4 4 4log 3 log 3 log 7 0.7925 0.7925 1.4037 0.1823

4log 3 0.7925Use and

4 4

7log 7 1.4037 log12

to find

4 4 4

7log log 7 log 1212

4 4log 7 log 3 4 4 4 4log 7 log 3 log 4

1.4037 0.7925 1 0.3888

Solve. Round to four decimal places.2 3 34 9x x We can’t unite bases! So give “log”2 3 3log4 log9x x 2 3 log4 3 log9x x

2 log4 3log4 log9 3log9x x

2 log4 log9 3log9 3log4x x

2log4 log9 3log9 3log4x

2log4 log9 3log9 3log4x

2log4 log92log4 log9 3log9 3lo

2g4

log4 log9x

3log9 3log2log4 log9

4x

4.2283x

1Pr logloga

b

ove ba

lo. . gal h s b

.1log

.b a

r h s

We change L.H.S to base “b”1log

loga

b

ba

loglog

b

b

ba

Challenge Evaluate 3 3

2 5log 5 log 2

2 53log 5 3log 2

2 53 3 log 5 log 2

2 5log 5 log 29 199

8-7 Natural Logarithms

Remember! ln xe x ln xe xln 210Evaluate eln 210 e

10 2 8

5ln6 8Solve x First isolate the “ln” then give it base “e”5ln6 85 5

x

8ln65

x 8

ln6 5xe e

856

6 6x e

0.8255x

ln(6 3) 3 10Solve x First isolate the “ln” then give it base “e”ln(6 3) 7x

ln(6 3) 7xe e 76 3x e 76 3x e 7 36

ex 183.2722x

53 1 10xSolve e 53 10 1xe 53 93 3

xe

First isolate the “e” then “ln” both sides5 3xe 5n l 3l nxe 5 ln3x

ln35

x

0.2187x

24 5 1xSolve e 24 5 1xe

24 64 4

xe

First isolate the “e” then “ln” both sides2 3

2xe

2n n 32

l lxe

32 ln2

x

3ln22

x

0.2187x

2ln 5xSolve e 2ln 5xe 2 5x 5 2x 3x

Write each exponential in logarithmic form2xe “ln” both sidesln ln 2xe

ln 2x

Write each exponential in logarithmic form0.35x e “ln” both sides0.35ln lnx e

ln 0.35x

Write each logarithm in exponential formln 0.6742x “e” both sidesln 0.6742xe e0.6742x e

Write each logarithm in exponential formln 22 x “e” both sidesln 22 xe e

22 xe

Write each expression as a single logarithm4ln9 ln 274ln9 ln 2749ln27

ln 2435ln3

5ln3

Write each expression as a single logarithm17ln 5ln 22

7

51ln ln 22

7

51ln 22

7 5ln 2 2

2ln 2

2ln 2

Challenge Evaluate ln5

3 3log 24 log 8e ln5

3 3log 24 log 8 e

ln5

3

24log8

e

ln5

3log 3 e

1 5 6

Challenge Solve 5 5log 2 log 3 45 lnx x xe 52log35 4x

x x 2 43x x

x

2 43 1x x

x

2 3 4x x x 22 12x x x

2 12 0x x 3 4 0x x

3 4 0x x 3 4x x Check -3 5 5log 2 log 3 45 lnx x xe

5 5log 2 log 3( 3) 3 435 ln e undefinedCheck 4 5 5log 2 log 4 3(4 4) 45 ln e 8 8 true

7-1 Operations on Functions

3 1 3 14 64x xSolve We can unite bases!3 1 3(3 1)4 4x x

3 1 9 34 4x x Now bases are same!3 1 9 3x x

3 9 3 1x x 6 4x 6 46 6x

23

x

Compound Interest You deposited $700 into an account that pays an interest rate of 4.3% compounded monthly.How much will be in the account after 7 years?12n

7t 700P

1ntrA P

n

0.043r

1ntrA P

n

12 70.043700 112

A

$945.34A

Compound Interest You deposited $1000 into an account that pays an interest rate of 5% compounded quarterly.a) How much will be in the account after 5 years?4n

5t 1000P

1ntrA P

n

0.05r

1ntrA P

n

4 50.051000 14

A

$1282A

Compound Interest You deposited $1000 into an account that pays an annual rate of 5% compounded quarterly.b) How long it take until you have a $1500 in your account?1

ntrA Pn

1500A1000P

?t

1ntrA P

n

40.051500 1000 14

t

41500 1000 1.0125 t Divide both sides by 1000

41.5 1.0125 t

41.5 1.0125 t “log” both sides now4log1.5 log1.0125 t

log1.5 4 log1.0125t

log1.5 4 log1.0124log1.0125 4log1. 25

501

t

8.16t yrs

Divide both sides by 4log1.0125

1( ) 2 3xGraph f x

X Y -2 3.125 -1 3.25 0 3.5 1 4 2 5

Use MODE 7{ }Domain All real numbers

{ 3}Range y

: 3Asymptote y

int : 0, 3.5y ercept

1( ) 2 3xGraph f x

X Y -2 3.125 -1 3.25 0 3.5 1 4 2 5

1( ) 22

x

Graph f x

X Y -2 8 -1 4 0 2 1 1 2 0.5

Use MODE 7{ }Domain All real numbers

{ 0}Range y

: 0Asymptote y

int : 0, 2y ercept

X Y -2 8 -1 4 0 2 1 1 2 0.5

1( ) 22

x

Graph f x

{ 0}Domain x { }Range All real numbers

: 0x

2( ) logGraph f x xPoints:(1, 0)(2, 1)1 , 12

{ 0}Domain x { .}Range All real no

: 0Asymptote x

2( ) logGraph f x x

{ 0}Domain x { .}Range All real no

: 0Asymptote x

3( ) log 2Graph f x x Shift 2units upPoints:(1, 0)(3, 1)1 , 13

After shift:(1, 2)(3, 3)1 , 13

3( ) log 2Graph f x x

{ 0}Domain x

{ }Range All real numbers

: 0Asymptote x

X=2X=-3

2( ) log ( 1)Graph f x x Shift 1unit rightPoints:(1, 0)(2, 1)1 , 12

After shift:(2, 0)(3, 1) 1.5, 1

2( ) log ( 1)Graph f x x

:{ 1}Domain x

{ .}Range All real no

: 1Asymptote x

2( ) log ( 3) 1Graph f x x

Shift 3units left and 1 unit upPoints:(1, 0)(2, 1)1 , 12

After shift:(-2, 1)(-1, 2) 2.5,0

2( ) log ( 3) 1Graph f x x

:{ 3}Domain x

{ .}Range All real no

: 3Asymp x

X=-3

Write an exponential function whose graph passes through the points (0, 15) and (3, 12)xy ab015 ab 15 a

Now replace second point and also “a=15”312 15b312 1

5 55

1 1b 3

12 0.9315

b 15(0.93) xy

Write an exponential function whose graph passes through the points (0, 256) and (4, 81) xy ab0256 ab 256 a Now replace second point and also “a=256”481 256b

481 2556 5

62 2 6

b 481 3256 4

b

32564

x

y

Exponential growth with given rate: 1 ty a r A house was bought for $96,000 in the year 2000. The house appreciates at a rate 7%. 1) Write an exponential equation that models the price after t years. 1 ty a r 96000 1 0.07 ty

96000 1.07 ty

2) Find the price in the year 2003. 96000 1.07 ty

396000 1.07y 117604.128y

$117,604.128price will be