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Exponential & Logarithmic Function V53.4.29

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Club Educate PathumwilaiC.E.P.W.Pathumthani Thailand

Text of Exponential & Logarithmic Function V53.4.29

CONTENTTop Map Exponential Function 1 14 21 28 43 57 63 64 67

Exponential ExponentialLogarithmic Function

Logarithm LogarithmCharacter Mantissa Antilog Graph of Exponential & Logarithmic Function

Exponential Function

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C.E.P.W....

Exponential Function (Must ! Very important for Expo)am an = am+n am = amn an (am )n = amn b = b n an = a n a0 = 1 01 a n an

a, b, am , bn

a, b 0 m, n

(ab)n = am an

(a + b)n an + bn (a b)n an bn b0 a0 a0

Fear Factor

(a b)2 = a2 2ab + b2

a2 b2 = (a b)(a + b)

(a + b)3 = a3 + 3a2 b + 3ab2 + b3 (a b)3 = a3 3a2 b + 3ab2 b3 a3 + b3 = (a + b)(a2 ab + b2 ) a3 b3 = (a b)(a2 + ab + b2 ) (a + b + c)3 = a3 + b3 + c 3 + 2ab + 2bc + 2ac

1

Exponential Function

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C.E.P.W....

Problems : 1. 1.1

49n + 142n n 7n +28 n

1

1.2

53n +1 +52n +1 n 52n +1 +5n +1

1

2.

218 212 36 + 26 311 315

2

Exponential Function

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C.E.P.W....

3.

(Lead! )

3.1 2222

3.2 2 + 2 + 2 + 2

3.3 3 + 3 3 + 3

4.

4.1 100010 , 101000 , 100100

3

Exponential Function

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C.E.P.W....

4.2 23 , 24 , 42 , 434 3 3

5.

1. 2 , 3 , 5 2. 3 , 5 , 2 3. 5 , 3 , 24 3 3 4 3 4

(Hardcore : Make your Brain Brighter The Brain) 6.

6.1 2 + 32 + 2 + 32 + 2 + 2 + 32 2 + 2 + 3

6.2

m 2 +1

m

1m 2 +1+m+m m 2 +1 m 2 +1+mm 2 +1m

m 2 +1

m

+1

4

Exponential Function

7.

1+x a b +x a c

1

+

Club Educate Pathumwilai1+x b c +x b a 1

+

C.E.P.W....

1+x c a +x c b

1

8.

(2x)3 = (3y)3 = (4z)3 1 + 1 + 1 = 1 3 2 + 27y2 + 64z2 8x x y z

9.

k

3 =a

b

2

= k a2 b 2 a2 + b 2 a2 b 2 3 3 3 3 3

5

2 4 4

Exponential Function

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C.E.P.W....

2

(a + b) 2ab2

2

2

2

9

a + b = a + 2ab + b = (a + b) + 2ab a b = a 2ab + b2 2

(a + b) 2ab

a b a b

9

3

3

2

(a + b) + 2ab = a + b

2

a b = (a + b) 2ab

(a + b) 2ab = a b

a >

Problems

1. 5 + 26 2. 5 26

3.

2 8 + 215

6

Exponential Function

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C.E.P.W....

4.

2 3x 1 + 22x2 3x 2 18 + 85

5.

6.

5 21 =

x y x y

7.

32 24

8. a b a < b a b 10 + 84 = a + b a2 + b2

()

7

Exponential Function

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C.E.P.W....

9.

4 5 17 415

10.

183+5

10 2

8+35

10+18

()

Problems Equation of Trick Mark Rider (Check ) 1.x1 x1

= 6x

2. 2x 2 7x + 1 2x 2 9x + 4 = 1

8

3. 2x 1 + 3x 2 = 4x 3 + 5x 4

Exponential Function

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C.E.P.W....

4. 4x 2 10x + 19 4x 2 10x + 3 = 2

5. 2x 2 11x + 69 + 2x 2 11x + 14 = 11

6.

3(x2 7x) + 26 = 4x2 7x + 10

9

Exponential Function

7.

x x+48x = x43 () x+48+ x x4+ 3

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C.E.P.W....

8.

2x2 32x2 7x + 7 = 7x 3 ()

9.

x x 1 + 2x 2 x 1 2x 2 = 1

10.

1 x + 1 2x + 5 x 9 18x = 42 x

10

Exponential Function

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C.E.P.W....

11.

x + 1 = 3 6 + 2 2x

Homework : 1.

a p ,q ,r a 0 ,1 a q p

p+q

ar

a q q+r

a p

a r r+p

2. a 1 x R x a a + x = x ( )

11

Exponential Function

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C.E.P.W....

3.

218 + 216 36 + 26 311 + 315

4. x x + x = k k x 3 + x 3 = 27041 1

5.

5 6

2 15

2

(O-Net 51)

6. 18 + 2125 34 (O-Net 51)3 4

12

Exponential Function4

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C.E.P.W....

7.

144

83

2

18 2 6

1

(O-Net 50)

8. 1 2

2

2 + 8 1 + 2 2 82 3

3

(O-Net 50)

13

Exponential Function

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C.E.P.W....

Exponential Function

f = {(x, y) R R+|y = ax ; a > 0 a 1}

... 2. R3. 4. 1.

1 R+

1-1Must to Jum @ Kow Jai Naa

Expo

Expo Expo14

Exponential Function

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C.E.P.W....

Problems : 1. y = 2x 2. y = (sin 30)2 4. y = 22x 5. y = 1 3. y = 2x+1100000000 x

Equation of Exponential Function

= 0 0 0 1 Expo Function

Problems : 1. 2x = 4 2. 2x = 4 3. 3 + 8 x=16

3 3x

= 81 x (O-NET)

x=

4.

4xy = 128 32x+y = 81 y (PAT1 .. 52)

15

Exponential Function

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C.E.P.W....

Solve Top Map Service5. 6 + 5 = 6. 11 + 10 = 7. 12 11 = 8. 13 + 12 = 9. 8 7 = 10. 7 + 5 =

11. 3 2

5x1

= 3 + 2

10x2

x

12. 11 230 = 6 + 5x 2

x

13. 72x + 4 7x 5 = 0

16

14. 5(5x + 5x ) = 26

Exponential Function

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C.E.P.W....

15. 6(25x ) + 11(23x ) 3(2x ) = 25x+1

16. 3x

2 +2

3x

2 +1

9x+1 + 27 = 0 ( King)

17. 3 4x + 6x + 2 9x = 0

17

18. 4 32x + 9 22x = 13 6x

Exponential Function

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C.E.P.W....

19. 12x 2(3x ) 9(4x ) + 18 = 0

20.

x > 0 8x + 8 = 4x + 2x+3 x (PAT1 .. 52)1. [0,1) 2. [1,2] 3. [2,3) 4. [3,4)

18

Exponential Function

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C.E.P.W....

21.

89x +3 = 12843x

( ) 22.

32x+y 3x+2y = 72 3x 3y = 8

23.

23x 14 22x + 56 2x 64 = 0

19

Exponential Function

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C.E.P.W....

24.

x 3(30x ) 6(15x ) 3(6x ) + 6(3x ) + 2(5x ) 10x + 2x 2 = 0

25. (25)2xx

2 +1

+ (9)2xx

2 +1

= 34(15)2xx

2

26.

tan x tan x tan x cot

cot x . x tan x

..

= tan x

20

Exponential Function

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C.E.P.W....

ExponentialTrick! 1.

2.

am > an m < n 1

am > an m >

Ex.

85x +1 95x +1

85x+1 > 95x+1

>

8 < 1 5x + 1 < 0 9

9

8 5x+1

95x +1 95x +1

: 95x+1 Expo >1 98 5x+1

> 9

8 0

21

Exponential Function

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C.E.P.W....

Problems :

1. Set of the answer from 2

2 1 x +3x+5

0

22

2 2 2 2 5. 23x +2 523x 23x + 2x +1 ( King)

Exponential Function

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C.E.P.W....

6.

[a, b] [c, d] (a + b + c + d)2

x x2 22x + 9(x + 2)2x + 8x2 (x + 2)2x + 9x2 2x + 8x + 16

7.

(sin 0)8x 2 5x+10 (sin 0)7x 2 4x+8

23

Exponential Function

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C.E.P.W....

Homework : 1. (tan 40)8x

2 x1

> (tan 40)2x

2 2x+1

2. 5(2x+2 ) (0.8)(5x+2 )

3. 32x 4 3x+2 + 9 > 01

24

Exponential Function

Club Educate Pathumwilai2x7

C.E.P.W....

4.

a > 1 ax 2

< a27x

5.

3 + 7

2x 2 +3x+1

0 a 1}

f = {(x, y) R R+ | y = ax ; a > 0 a 1}

Log g = {(x, y) R+ R | y = log a x ; a > 0 a 1} Must to Jum @ Kow Jai Naa

y = loga x log x a

x = ay x y

Loga Loga28

Logarithmic Function

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C.E.P.W....

1. log a 1 = 0 : log 1 = 0 2. log a a = 1 : log = 1

Hua Kor Tor Pai Nee Taa Mai Dai Hmmm Dont Walk ago !

3. log a (mn) = log a m + log a n : log m n

4. log a = log a m log a n : log x 5. log b y ax = log b a :

Ex. log10 10 = log10 (5 2) = log 5 + log 2

log

log 5 = 1 log 2 log 2 = 1 log 5

log

6. alog a m = m : 7. alog b m = mlog b a : 8. log b a =log m b 1 log m a

y

9. log b a = : log a b

: m > 0 , 1 , 1 :

1.

(Common Logarithms) log Ex. log10 2 = log 2

log log

2.

(Natural Logarithms) log

e 2.7182818

Part Part Here we go !!!

ln x loge x 9

29

Logarithmic Function

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C.E.P.W....

Problems1.

log (Take log)1.1 23 = 8

1.2 53 = 125 1.3 33 = 27 1.4 43 = 64 1.5 pq = r 2.

log Exponential ( )2.1 log 5 25 = 2 2.2 log 3 27 = 3

2.3 log 1000 = 3 2.4 p = log q r

3.

7log 7 52 + 5 log 2 43 2 log 9 33

30

Logarithmic Function

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C.E.P.W....

4.

81

1 log 5 3

+ 27

log 9 36

+3