HEG Maths 1 Partie 2 PDF

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Summary

Support de cours, partie 2, 2021-2022.
Comprends les chapitres des fonctions exponentielles et logarithmes...

Description

an

n∈N a

n

n facteurs

n

an · am = an+m

(an )m = anm

( a · b) n = a n · bn

 an

an = an−m am √ n

a0 = 1

am =

p √ n m

m √ n a

a= m

an = n=2

a

a | · a ·{z. . . · a}

√

a

√

nm

√ n

a

am

b

=

1 = a−n an √ n

ab =

√

np

an bn

a1/n =

a

√ √ n n a b

amp =

√ n

am

a>0 a

4

√ n

√

4=2

a4 · a28

x25 · x−14

u−22 · u7

b4n−7 · b−n+7

h−4 h−7

n0 n−5

z 3−6n

(−a)17 (−a)15

Cx C

ex e−x

54a3 x5 y 10 18a−4 xy 8

(−2a)5

(−x6 )4

(−10x)6 (−100x)3

(ab0 c−3 )

(−3x4 )3

(a−3 b5 )3

(a5 b−8 x)−4

(−a)4n+3 (−an )4

(4a3 b7 )5 8(a2 b3 )2

(2x2 )1/2 √ 6 8 · x−1



√4

27

√ 3

√4

b8

 √ 15 5 a6

√ 2n−1

a

a4n−2

a5 b6

−4

x4 √ 5 x7

an

√ n √

√ n·m n ap

1 a2n2 −n

x8n

a 6= 1 f : R −→ R>0 x 7−→ f (x) = ax .

tx−4 · t6−x z5



a4 b−4 c7

1 √ 3 √ 2n x4 √ 4n x3 ((xn )3 )

2

−n

3

ax

x

ax a1

a = 12 a=3 a = −4 x =

x a=1

1 2

f (x) = 1x = 1

y=1

limx→+∞ f (x) = +∞

limx→−∞ f (x) = 0 f

a

f (x) = ax P (3; 10,648)

7x+6 = 73x−4 35x−8 = 9x+2 ; 2

32x+3 = 3(x ) 2−100x = 0,5x−4  6−x 1 = 5. 5

a0 \ {1} x

80%

a

x ∈ R>0

loga x y = loga x

loga a = 1

loga 1 = 0

⇐⇒

ay = x.

loga ax = x

11

alog a x = x

= ...

log10

1 10

= ...

log10 1000 = . . .

log10

1 √ 4 10

= ...

log10 107

√ log10 3 100 = . . .

log10 1

= ...

log2 8

= ...

log2

√5

8

= ...

log2 64

= ...

log3

√4

27

= ...

log2 1024

= ...

log4

√5

64

= ...

1 log2 512

= ...

log11 11− 5 = . . .

log3 729

= ...

log4

= ...

log2

log3

√ 4

3

log10 (−1000) = . . .

√3

16

= ...

1 √ 3 32

= ...

log4 0

= ...

x

= log2 32

log2 x

= 4

x

= log5 125

log5 x

= 5

log11 x = 4

loga x

4

8log8 32 = x

log4 x

= 3

logx 1

= 0

logx 125

= 3

log4 x

= −3

logx 1000 = 3

log2 x

= 0

log4 128

log27 81 = x.

= x

a ∈ R>0

f

f : R>0 −→ R x 7−→ f (x) = loga(x).

x

a=2

f (x) = log2 (x) x 7−→ f (x) 2 4 8 16 1 1/2 1/4 1/8

7−→ 7−→ 7−→ 7−→ 7−→ 7−→ 7−→ 7−→

1 2 3 4 0 −1 −2 −3 y = log2 (x)

lim log2 (x) = −∞ x→0

u v x ∈ R>0

lim log2 (x) = ∞

x→∞

p∈R

• loga (u · v) = loga u + loga v

• loga

u  v

= loga u − loga v

• loga xp = p · loga x

α · log a + β · log b √  log a5 · b 

log10

♣= ♣=



log

1000 · a8 a 2 · b7



a 7 · b4 √ √ 5 3 a2 · b





an · bm−4 log an−2 · bm−6



log

√ 6

a 2 · b7 √ a · b4

!

α · log a + β · log b + γ !   0,001 · a8 · b5 (10 · a · b)3 log10 log10 100 · a−2 · b3 100a4 b−4



loga(♣) = loga()

loga(♣) = loga()

♣=

⇐⇒

loga(♣) = loga()

log x =

1 · log 20 − log 2 2

log x =

1 1 · log 9 + · log 8 3 2

log x = log

3 4 5 − log + log 4 2 3

log x = 3 · log a + 7 · log b log(3x − 4) = 2 · log 3 log(x + 1) + log(x + 2) = log(5x + 5)

log x := log10 x log x = 1 + log 6 log(3x + 1) = 3 log x =

1 − log 5 − log 8 2

log 2x = 2 · log x + 1

log10 589400000

loga x logn x y = logn x

log10 5894 ∼ = 3,77 p log10 58,94 log10 5,894

x

a

n loga x

x = ny loga x = loga ny = y · loga n = logn x · loga n.

logn x =

loga x loga n

logn x =

log10 x log10 n

log10

log7 200

log5,1 34,7

log25 125

n  1 . S(n) = 1 + n

log49 2401

logn x

S(1) S(2) S(12) S(52) S(365) S(1000) S(10′ 000) S(105 ) S(106 ) S(107 ) S(108 ) S(109 )

= = = = = = = = = = = =

2  2 1 + 12 = 2,25   1 12 ∼ 1 + 12 = 2,613   1 52 ∼ = 2,693 1 + 52  365 1 ∼ 2,714 1 + 365 = 2,71692393224 2,71814592683 2,71826823717 2,71828046932 2,71828169255 2,71828181487 2,71828182710. S(n)

n

e

e

π  n 1 = 2,718281828459045 . . . 1+ n→∞ n

e = lim

x

e

x

ln x := loge x

ln e

ln 1

ln

√

15

e

1 ln √ e9

ax = b x= 2,76x = 12000

N (t) N0 = 200

t

log10 b ln b . = log10 a ln a

47,3x = 8653,37

100 · 3x = 5000

N0 N (1) N (2)

N (t) 109

C

t

n

C(n) C = 3500

C(20)

t = 2,5% 10000 1,75 %

15000 3,8 %

V 0,03t

V (t) = V0 e

t

t 400000

7 650000

2800 5160 35000

8% 15000

t P (t) = 1 − e−0,31t

95%

%

%

11%

3′ 545′ 247 4′ 458′ 069

5

V (t)

t  V (t) = 2600 1 − 0,51 ·

−0,075t

A(t)

3

.

t

A(t) =

36 1 + 200

−0,2t

10

.

N (t) t N (t) = t

13 1 + 16 · 2,5−0,36t

N (t)

172,00 131,00

264,80 236,05

N (t)

N (t) =

t 800 · e0,36·t 399 + e0,36·t

N (t) N (t) =

625

800 . 1 + 399 · e−0,36·t

416,79

C(t)

P (t) E(t) = C(t)/P (t) 0,346

P (t)

P (t) =

1 1,25 + 7 · e−0,06t

t

50%

2,25%

3%

1,8%

3

53,03 47,43

60,00 57,02

P (t) P (t) =

1 1,5 + 8 · e−0,02t

0 ≤ P (t) ≤ 1 t 50%

N (t)

t

N (t) =

40500 45 + 2 ln(t)2

2,24% 61362,65 A D

t A=

n 1−

t  1 n · D 1+t

4,5% 9411,90

a32 x11 u−15 t2 b3n h3 n5 z 2+6n a2 C x−1 e2x 3a7 x4 y2

a12 b−12 c−21

−32a5 x24 −x3 a−n c3n −27x12 a−9 b15 a−20 b32 x−4 −a3 128a11 b29 33/4 an/3 x13/5 3−1/2 b2 a18 a1−2n x4n a2 apm x4n 1

x · |x| a−20 b24

a3 = 10,648

a=

√ 3 10,648 = 2,2

f

f (x) = 2,2x  x+2 x + 6 = 3x − 4 x=5 35x−8 = 32x+4 35x−8 = 32 2 2 5x − 8 = 2x + 4 x=4 (x − 3)(x + 1) = 0 x = −1 x = 2x + 3 x − 2x − 3 = 0  −1 x−4 −100x −x+4 −100x −100x = −x + 4 x = −4/99 x=3 2 =2 2 = 2  −1 6−x x−6 1 1 x−6=1 x=7 5 =5 5 =5 ∼ 3117 N (4) = 5400 N (1) ∼ = 1039 N (2) = 1800 N (3) = C(n)

C(n) = 1,04n · 5000

n

= 8005,15 C(12) ∼ E(n) ∼ 230 E(8) = ∼ 179 ∼ 128 E(5) = E(12) = ∼ 3,28 q(5) = 10 · 0,85 =

t=7

log10 1 = 0 log10

−9 log11 11 log4

− 45

1 10

E(n) = 0,92n · 350

n

t

= −1 log10 1000 = 3 log10

10 · 0,8t = 2 1 √ 4 10

= − 14 log10 107 = 7 log10

√ 3 100 =

√ √ √ 5 4 5 log2 8 = 3 log2 8 = 35 log2 64 = 6 log3 27 = 34 log2 1024 = 10 log4 64 = 53 log2 √ √ = − 45 log3 729 = 6 log4 3 16 = 32 log3 4 3 = 14 log2 √31 = − 53 log10 (−1000)

2 3 1 512

=

32

log2 x = 4 ⇒ x = 24 = 16 x = log2 32 = 5 x = log5 125 = 3 log5 x = 5 ⇒x= = 3125 log11 x = 4 ⇒ x = 114 = 14641 8log8 32 = x ⇒ x = 32 log4 x = 3 ⇒ x = 43 = 64 logx 125 = 3 ⇒ x3 = 125 ⇒ x = 5 log4 x = −3 ⇒ x = 4−3 = 164 logx 1 = 0 ⇒ x ∈ R>0 logx 1000 = 3 ⇒ x = 10 log2 x = 0 ⇒ x = 1 log4 128 = x ⇒ x = 27 log27 81 = x ⇒ x = 34 55

t = loga u s = loga v r = loga x u = at t−s loga(u·v) = s+t = log a u +loga v v as = a p xp = (ar ) = ap·r loga xp = p · r = p · log a x

33 11 ·log b 5 ·log a+ 3

 √  1 log a5 · b = log a5 + log b 2 = 5 log a + 12 log b  n m−4    a b = log a2 b2 = 2 log a+2 log b log an−2 bm−6 log10

3 3

7

10b 1000a b log10 100a 4 b−4 = log10 a −5 + 10 log10 a + 2 log10 b

1000a8 a2 b7

= log10

1000a6 b7

loga uv

u · v = at · as = as+t = t−s = loga u−log a v

 74   33 11  log 5√a 2b 3√ = log a 5 b 3 = a b √ 6 2 7 a ·b √ = − 16 log a− 17 log 6 log b a·b4 (10ab)3

log10 = 3 + 6 · log10 a − 7 · log10 b 4 b−4 =  −5 100a  0,001·a8 ·b5 10 = 1 + 7 · log 10 b − log10 a log10 100·a−2 ·b3 = log10 10 · a · b2 =

1 · log 20 − log 2 2 √ log x = log 20 − log 2 √ 20 log x = log 2 log x =

x=

√ √ 4· 5 √ = 5 2 1 1 · log 9 + · log 8 2 3 log x = log 3 + log 2 log x =

log x = log 6 x = 6 3 5 4 − log + log 4 3 2   3 3 5 · · log x = log 2 4 4 45 log x = log 32 45 x = 32 log x = log

log x = 3 · log a + 7 · log b log x = log a3 + log b7 log x = log a3 b7 x = a3 b7 log(3x − 4) = 2 · log 3

log(3x − 4) = log 9 3x − 4 = 9 3x = 13 13 x = 3

log(x + 1) + log(x + 2) = log(5x + 5) log [(x + 1) · (x + 2)] = log(5x + 5) x2 + 3x + 2 = 5x + 5 x2 − 2x − 3 = 0

(x − 3)(x + 1) = 0 x=3

x = −1

log x = 1 + log 6 log x = log 10 + log 6 log x = log 60 x = 60 log(3x + 1) = 3 log(3x + 1) = log 1000 3x + 1 = 1000 3x = 999 x = 333 log x

=

log x

=

log x

=

x =

√ 10 40

1 − log 5 − log 8 2 √ log 10 − log 40 √ 10 log 40

log 2x = 2 · log x + 1 log 2x = log x2 + log 10 log 2x = log(10x2 ) 2x − 10x2 = 0

2x · (1 − 5x) = 0 x=

x=0

1 5

log10 589400000 = log 10 (5894 · 100000) = log 10 5894 + log10 100000 = 3,77 + 5 = 8,77 5894 log10 58,94 = log 10 = log10 5894 − log 10 100 = 3,77 − 2 = 1,77  100  p 0,77 5894 1 1 log10 5,894 = = 0,385 · log10 = · [log10 5894 − log 10 1000] = 2 2 2 1000 log7 200 =

log10 200 = 2,723, log10 7

log25 125 =

ln e = 1,

log 5,1 34,7 =

log10 34,7 = 2,177 log10 5,1

log5 125 3 log7 2401 4 = log49 2401 = = = 2. 2 log5 25 2 log7 49

ln 1 = 0,

√ 15 15 15 ln e = , · ln e = 2 2

9 1 ln √ = − . 9 2 e

x = 9,251 x = 2,351 x = 3,561 t N (t) = 2t · N0 .

2t · 200 = 109

2t = 5000000 log(2t ) = log 5000000

t · log 2 = log 5000000 log 5000000 ∼ t = = 22,25. log 2

C(20) = 1,02520 · 3500 = 5735,15 n 24e n

10000 · 1,0175n = 15000 C ·1,038n = 2·C 3 ∼ 29,45 ∼ = 30 = n = loglog 1,038

V (0) =

V0 = 400000

n=

∼ 23,37 =

log 2 log 1,038

∼ 18,6 = 4 ∼ 37,17 ∼ = 38 = n = loglog1,038

400000

∼ 493471 V (7) = 400000·e0,21 =

400000 · e0,03t = 650000 e0,03t = 1,625

0,03t = ln 1,625 ln 1,625 ∼ t = = 16,18. 0,03

N (t)

t

t=0 N (0) =

log 1,5 log 1,0175

  V0 e0 = V0 

V (t) = 400000·e0,03t

n=



C · a0 = C 2800

N (t) = C · at

N (t) = 2800 · at

C = 2800

N (5) = 2800 · a5 = 5160. a=



5160 2800

1/5

= 1,13.

∼ 17524 N (15) = 2800 · 1,1315 =

. A(n)

n A(n) = 35000 · 0,92n . n

35000 · 0,92n = 15000 n=

t

log 15 − log 35 ∼ = 10,16. log 0,92

P (t) = 0,95 1 − e−0,31t = 0,95 0,05 = e−0,31t ln 0,05 = −0,31t ln 0,05 ∼ t = = 9,66. −0,31

2005 + t

A(t) = A0 · at A(0) = A0 = 3000  1/10 ∼ = 1,052 a = 35

B(0) = B0 = 6000  1/10 ∼ = 0,96 a = 23

t

B(t) = B0 · bt A(t) = 3000 · at B(t) = 6000 · bt

A(t) = B(t)

3000 · 1,052t = 6000 · 0,96t t  1,052 = 2 0,96   1,052 t · log = log 2 0,96 log 2  = 7,57.  t = log 1,052 0,96

A(10) = 3000 · a10 = 5000 5,2% B(10) = 6000 · b10 = 4000 4%

F (t) = F0 · f t

F (5) = 12212 ·

f5

= 8962

2010 + t

F (t) = 12212 · f t F (0) = F0 = 12212  8962 1/5 ∼ f = 12212 = 0,94 6% M (t) = 4500 · mt

M (0) = M0 = 4500  6924 1/5 ∼ 1,09 m = 4500 =

9%

t

M (t) = M0 · mt

M (5) = 4500 · m5 = 6924

F (t) = M (t) 12212 · 0,94t = 4500 · 1,09t t  0,94 4500 = 1,09 12212     4500 0,94 = log t · log 1,09 12212  4500  log 12212   = 6,74. t = log 0,94 1,09 M (t)

P (t)

t t M (t) = · M (0) 41880 ∼ M (0) = = 84250 0,896

6

M (6) = 41880

· M (0) = 41880

P (6) = a6 · 52000 = 31530 a=



31530 52000

 16

= 0, 92. 8%

t

P (t) = M (t) 0,92t · 52000 = 0,89t · 84250   0,92 t 84250 = 52000 0,89   0,92 84250 t · log = log 52000 0,89 log 84250 52000 ∼ t = = 14,5. 0,92 log 0,89

V (t)

2000 + t

V (0) = 3545247 V (15) = 3545247 · a5 = 4458069

V (t) = 3545247 · at a=



4458069 3545247

1

15

V (15) = 4458069

∼ 1,01539. =

∼ 5193600. V (25) = 3545247 · 1,0153925 = 5000000 3545247 5000000 t · log 1, 01539 = log 3545247 log 5000000 3545247 ∼ t = = 22,512. log 1,01539 1,01539t =

= 305,9 V (0) = 2600 · 0,493 ∼ t

V (t) = 1800  3 = 1800 2600 · 1 − 0,51 · e−0,075t   9 3 = 1 − 0,51 · e−0,075t 13 r 9 3 −0,075t −0,51 · e = −1 13 0,1154 e−0,075t = 0,51 = 19,82. t ∼

V (10) = t

V (t) = 15

15000 36 = 15 1 + 200 · e−0,2t 36 = 15 + 3000 · e−0,2t 21 = 3000 · e−0,2t 0,007 = e−0,2t

ln 0,007 = −0,2t ln 0,007 t = =− = 24,8 0,2

36 1+200·e−2

∼ 1,28 =

A(t) = 172 · at

2005 + t

V (t) = 264,8 · v t

A(11) = 264,8 = 172 · a11 a=



264,8 172

1

11

= 1,04 = 104%

4% V (11) = 236,05 = 131 · v 11 v=



236,05 131

1

11

= 1,055 = 105,5%

5,5% A(8) = 172 · 1,048 = 235,4 t

V (8) = 131 · 1,0558 = 201,05

A(t) = V (t) 172 · 1,04t = 131 · 1,055t 1,055t 172 t 131 = 1,04 t 1,055 172 1,04 131 =    172  log 131 = t · log 1,055 1,04

2025

N (0) =

800 399+1

 172 log 131 ∼ 19,02.  = t= 1,055 log 1,04

= 2

N (15) =

800 · e0,36·15 ∼ = 285,5 399 + e0,36·15 e−0,36·t

800 · e0,36·t e−0,36·t 800 · e0,36·t e−0,36·t 800 800 · e0 = = = · −0,36·t −0,36·t 0,36·t −0,36·t 0,36·t −0,36·t 0 1 + 399 · e−0,36·t e 399 · e +e 399 · e +e e 399 + e t→∞

e−0,36·t → 0 lim N (t) =

t→∞

800 = 800 1+0

1 + 399 · e−0,36·t 800 = 625 · −0,36·t 1 + 399 · e 625 1,28 = 1 + 399 · e−0,36·t −1 0,28 = 399 · e−0,36·t : 399 0,28 −0,36·t = e ln( ...)  399 ln

0,28 399

= −0,36t ln

t=



0,28 399

−0,36



∼ = 20,17

21e 5,5% 8,76%

3%

∼ 41,54% P (30) = 32886,50

80%

2,31% 2,5%

P (50) ∼ = 22,51% N (15) ∼ = 678

11632,70

38

3,75%

63,04

61,37

2/3...