G 8 ,H و ي+95 ا . ر ا 4+5 )ع - ا / ت د 5 ا EF

(1)

ر ا +,1 ا ت D & ا / /0 ا ر ت C و ت & ا+ و ت د 5 EF / /0 ا ت

G 8 ,H و ي+95 ا . ر ا 4+5 )ع - ا / ت د 5 ا EF

10 x = = = = a

( . 7L ? و ! !"ا ر ا ت 7- ا / /0 ا ي & لاود E و !ارد / /0 ا 6 $

لاود . ر

= 0 ا ب & ا 0 ا و ء O L ا ت ء F ا و ء AFPا ضر"ا و ة & ا م $

د A S ا م 5 ا +,1 ا ت D & ا

ي+ ,- ا . ر ا اد ي+ ,- ا . ر ا اد !ارد ى+<أ د $ا ت 7 ا ر ا 8 9 ا

ر ا اد س !: .

a ي+95 ا . ر ا اد . ر ا اد !ارد س !:

a

س

ت $ ا د $

10 :

@ @

--20201111 201

2010

@@ @@ @@ @@

0

ي+ ,- ا . ر ا اد / /0 ا ي+ ,- ا . ر ا ا +,1 ا ت 2 3 ا 4+5

ي+ ,- ا . ر ا اد !ارد / /0 ا 6 $ ف+5 ا

! !"ا . ر ا ت 7

ا ا 8 9 و ا ر ا 8 9 ا 4+5

(((( ))))

x ֏֏֏֏ln u x

ا 2أ اد & / /0 ا

(((( )))) (((( ))))

x u' x u x

֏

س !: . ر ا اد 6 $ ف+5 ا .

a

/ ى+<أ دا =4 ر ا لاو ا > ?

@A3 ا دا .









س . . ي

(2)

N



I

Fonction logarithme népérien

ا ا

x 1 x

֏

֏֏

ل 1 ا 6 $ A

֏

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

ل 1 ا 6 $ 2أ لاود E,8 نذإ ،

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

،

=4 م 5- ة Fو 2أ اد E,8 و .

1

اد ا 2"ا ا ا =ه ي+ ,- ا . ر ا

x 1 x

֏

ل 1 ا 6 $

֏

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

=4 م 5- = ا و .

1

O + D 7 O + و :

.

ln

ا ا > +5 $ 1 ل 1 ا =ه

ln

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

.

ln 1 ====0

ا ا V D S

ln

ل 1 ا 6 $ ق 8

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- و

]]]] [[[[ (((( ))))

¹

:

x 0 , , ln x x

∀ ∈ +∞ ′′′′ =

∀ ∈∀ ∈ +∞+∞ ==

∀ ∈ +∞ =

ا ا 6 $ 5HS اO

ln

ل 1 ا

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

. يأ

]]]] [[[[ :

a ,b 0 , , a b ln a ln b

∀ ∈ +∞ < ⇔ <

.

]]]] [[[[

a ,b 0 , , ln a ln b a b

∀ ∈ +∞ = ⇔ =

∀∀ ∈∈ +∞+∞ == ⇔⇔ ==

∀ ∈ +∞ = ⇔ =

.

ln x ====0 ⇔⇔⇔⇔x ====1

و

ln x >>>>0 ⇔⇔⇔⇔x >>>>1

و

ln x <<<<0 ⇔⇔⇔⇔0 <<<<x <<<<1

.

E0 و

a

ل 1 ا /

b

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- ،

(((( )))) :

ln ab ====ln a++++ln b

XY

(((( )))) (((( )))) :

x 0 , F x ln kx

∀ > =

(((( )))) و

x 0 ,u x kx

∀ > =

Z F ،

k∈∈∈∈ℝℝℝℝ₊₊₊₊^∗^∗^∗^∗

.

-

(((( )))) (((( )))) :

x 0 , F x ln u x

∀ > =

∀ >∀ > ==

∀ > =

نذإ ل 1 ا E$ ق 8 V D S

F

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- و :

(((( )))) (((( (((( )))) )))) (((( ))))

¹

x 0 , F x ln u x u x

′ ′ ′ x

′ ′ ′

∀ > = × =

∀ >∀ > == ×× ==

∀ > = × =

(((( )))) نذإ

c / F x ln x c

∃ ∈ = +

∃ ∈ℝℝℝℝ = +

E أ / و

x ====1

،

ln k ====c

.

G- و

(((( ))))

ln kx ====ln x++++ln k

.

-5[و اذإ :

x ====a

و

k ====b

ن\4

(((( )))) :

ln ab ====ln a++++ln b

.

ˆbn ÿa

ˆbn ÿa Z

Z @@

Ùi@óïäbrÜa@óå Üa Ùi@óïäbrÜa@óå Üa

@@--20201111 201

2010@@@@@@@@ 0

óïámŠbÈíÜÜaßaì‡Üa

óïámŠbÈíÜÜaßaì‡Üa ^@ ^@

ârïéÜaæia@óïÝïèdnÜa@óîíäbrÜa^óibïä

@ãíÝÈ@bîŠíÜb

ﺌ

ŒïÐ

óï bî H‘î†cI



א

א

אא

ا لاو ا > +5 $ 1 د F :

f : x x

ln x

֏

֏֏

֏

(((( ))))

g : x ֏֏֏֏ x −−−−1 ln x

h : x ֏֏֏֏ ln ln x

.

=4 EF

ℝℝ ℝℝ

ا ت د 5 ا :

((((

²

))))

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

(((( )))) (((( ))))

ln x −−−−1 −−−−ln 3 x++++1 ====0

2

x 1

ln 0

x 1

+ ++

 + 

 

=

==

 =

 

+ + +

 + 

 

=4 EF

ℝ ℝ ℝ

ا ت & ا+ ا

ℝ

:

(((( ))))

ln 3 x++++2 <<<<0

(((( )))) (((( ))))

ln x −−−−1 <<<<ln 2 x−−−−1

2

x 1

ln 0

x 1

−

−−

 − 

 

≥

 ≥

 

++ ++

 

Definition

I^I





     

   

 

 

    

/ هر,

(3)

نأ X + D / ,

(((( )))) :

i n i n

i i

i 1 i 1

ln a ln a

= =

== ==

= =

 

 

 

=

 =

 

 

 

 



∏ ∏ ∏ ∏



∑ ∑ ∑ ∑

Z F

a1 2

و و

a

...

n

و 5HS , 8 8F ادا $أ

a

.

E0 و

a

ل 1 ا /

b

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

و ، E0 /

r ℚ ℚ ℚ

- ،

ℚ

:

1

(

ln ln a

a

 

 = −= −= −= −

  

 

2

a

(

ln ln a ln b b

 

 ==== −−−−

  

 

3 (((( ))))

^r

(

ln a ====r ln a

4

1

(

ln a ln a

= 2

==

=

1 ( -

1 1

:

a 0 ,0 ln 1 ln a ln a ln

a a

 

∀ > = = × = +

∀ > = =  × = +

 

G- و

ln 1 ln a a

 

 = −= −= −= −

 

  

 

.

 

2 ( -

a 1 1

:

ln ln a ln a ln ln a ln b

b b b

   

= × = + = −

   

   

E0 و

a

/

b

∗

∗ + + +

ℝ+

ℝ ℝℝ

.

3 ( ن آ اذإ

r ====n∈∈∈∈ℕℕℕℕ

1 2 n

:

a ====a ====....====a ====a

6 $ EA& و :

ln an ====n ln a

ن آ اذإ

r = − ∈= − ∈= − ∈= − ∈n ℤℤℤℤ⁻⁻⁻⁻

:

r n n

n

a 0 , ln a ln 1 ln a n ln a r ln a

∀ > = a = − = − =

∀ > = = − = − =

ن آ اذإ

r p

= q

=

:

=

r qr p

a ,q ln a ln a ln a p ln a

∀ > = = =

نذإ

a 0 , ln ar r ln a

∀ > =

آ اذإ ن

xy >>>>0

ن\4

(((( )))) :

ln xy ====ln x ++++ln y

و

ln x ln x ln y y

 

= −

 

 

و

ln x2 ====2 ln x

E0 /

x

ل 1 ا

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

E0 و /

n

∗

∗∗

ℕ∗

ℕ ℕ

- ،

ℕ

:

n 1

ln x ln x

= n

=

.

=

د 5 ا b D :

A ====ln 2 −−−− 2 ++++ln 2++++ 2

.

(((( ))))

²⁰⁰⁹

(((( ))))

²⁰⁰⁹

B ====ln 2 −−−−1 ++++ln 2 ++++1

=4 EF

ℝ ℝ ℝ

د 5 ا

ℝ

(((( )))) :

ln − +− +− +− +x 3 ====2

& ا+ ا .c

(((( )))) :

ln − +− +− +− +x 3 ≥≥≥≥2

II (

ا ا > +5 $ 1 ل 1 ا =ه

ln

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

.

1 (

د -ا ت .

xlim ln x

→+∞

→+∞→+∞

→+∞ = +∞= +∞= +∞= +∞

و

x 0

lim ln x

+ + +

→ +

→→

→ = −∞= −∞= −∞= −∞

و

x

lim ln x 0 x

→+∞

→+∞→+∞

→+∞ ====

-

x

:

x 0 x 0

lim ln x lim ln1 lim ln x

+ + x

+ +

+ + →+∞→+∞→+∞→+∞

→ →

→→ →→

→ →

= − = − = −∞

= −= − = −= − = −∞= −∞

= − = − = −∞

.

   

     

/ هر,





(4)

ˆbn ÿa ˆbn ÿa Z

Z @ @

Ùi@óïäbrÜa@óå Üa Ùi@óïäbrÜa@óå Üa

@ @

--20201111 201

2010

@@ @@ @@ @@

0

óibïä

ârïéÜaæia@óïÝïèdnÜa@óîíäbrÜa

@ãíÝÈ@bîŠíÜb

ﺌ ŒïÐ

óï bî H‘î†cI



א

א

אא

2 ( ا0 ا تا 1 لو0

ln

ا ا ل 1 ا 6 $ ق 8 V D S

ln

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- و

(((( )))) (((( ))))

¹

:

x 0 , ln x 0

x

∀ > ′′′′ = >

∀ >∀ > == >>

∀ > = >

+∞

+∞+∞

+∞

1

0

x

+ ++

+

+ + +

+

(((( )))) (((( ))))

^ln ^′′′′ ^x

+∞

+∞+∞

+∞

0

−∞

−∞−∞

−∞

ln x

3 ( ا0 ا 5 6 7 . + ا عو 9 ا

ln

-

x 0

:

lim ln x

++ ++

→

= −∞= −∞

ا ا 6-&- يد $ بر 8 d ار"ا ر & نذإ .

ln

xlim ln x

→+∞→+∞

→+∞→+∞ = +∞= +∞= +∞= +∞

x

و

lim ln x 0 x

→+∞→+∞→+∞

→+∞ ====

نذإ 6-&- E 2 4"ا ر & Gه 1 ا 1 $+4 E,8

ln

را 1D

+∞+∞

4 ( د0 ا

e

ا ا ل 1 ا 6 $ 5HS اO و A

ln

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

نذإ

]]]] [[[[ :

(((( )))) ]]]] [[[[

x 0 x

ln 0 , lim ln x , lim ln x ,

+ + +

+ →+∞→+∞→+∞→+∞

→→→

→

 

+∞ = = −∞ +∞

+∞ = = −∞ +∞

.

نأ D و

]]]] [[[[

1∈ −∞ +∞∈ −∞ +∞∈ −∞ +∞∈ −∞ +∞,

د 5 ا ن\4

ln x ====1

ل 1 ا =4 ا F و VF E,8

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

ف+& D G O + .

e

- و :

ln e ====1

و

e∉∉∉∉ℚℚℚℚ

و

e ≈≈≈≈2.718

.

(((( ))))

^k

x 0 , k , ln e k

∀ > ∀ ∈ =

∀ > ∀ ∈ℚℚℚℚ =

و

x 0 , k , ln x k x ek

∀ > ∀ ∈ = ⇔ =

∀ > ∀ ∈ℚℚℚℚ = ⇔ =

.

x 0 , k , ln x k x ek

∀ > ∀ ∈ ≤ ⇔ ≤

∀ > ∀ ∈ℚℚℚℚ ≤ ⇔ ≤

5 ( ا0 ا 5 6

ln

ا ا 6-&- س د 5 7 A4أ = ا H8- ا =4

ln

=ه

1

:

y ====x−−−−1

.

ا ا 6-&- س د 5 7 A4أ = ا H8- ا =4

ln

=ه

e 1

:

y x

=e

=

==

.

O y

1 1

1

y=lnx

e

y ====x−−−−1

y 1x

=e

==

=

óïámŠbÈíÜÜaßaì‡Üa N óïámŠbÈíÜÜaßaì‡Üa

@ @

(5)

/ + 01

/0

= D 4+5 ا د 5 ا ا ا

f

(((( )))) :

f x ====x−−−−ln x

1 ( د F

Df

تا & -$ ت 7- ا d Fأ .c

Df

.

2 ( ا ا تا+ f سردأ

f

3 ( ا ا 6-&- g 7 V ا عو+L ا سردأ .

f

4 ( ا ا 6-&- h9 أ .C- 5 . 5 =4

f

III (

x 0

lim x ln x 0

+ + +

→ +

→→

→ ====

و

x 1

lim ln x 1 x 1

→

→ ====

−

و

−

(((( ))))

x 0

ln 1 x

lim 1

x

→

+ + + + ====

n x 0

n , lim x ln x 0

+ ++ +

∗∗

→→

∀ ∈ =

∀ ∈ℕℕℕℕ =

و

x n

n , lim ln x 0 x

∗

∗∗

∗

→+∞

∀ ∈ →+∞ =

∀ ∈ =

∀ ∈ℕℕℕℕ =

x 0 x 0 x 0 t

ln 1

1 x ln t

lim x ln x lim x ln lim lim

x 1 t

x

+ + +

+ + + →+∞→+∞→+∞→+∞

→ → →

→→ →→ →→

→ → →

 

 

 

 

 

 

 

   

= − = − = −

= −= − = −= − = −= −

= −  = − = −

 

.

(((( ))))

x 1

lim ln x ln 1 1 x 1

→

= ′′′′ =

= =

−−

.

(((( ))))

x 0

ln 1 x

lim ln 1 1

x

→

++ ++

= ′′′′ =

= =

.

(((( ))))

n n n

x 0 x 0 t 0

1 1

n , lim x ln x lim x ln x lim t ln t 0

n n

+ + +

∗

∗∗

∗

→ → →

∀ ∈ = = =

∀ ∈ℕℕℕℕ = = =

.

n

n n

x x t

ln x 1 ln x 1 ln t

n , lim lim lim 0

x n x n t

∗

∗∗

∗

→+∞ →+∞ →+∞

 

∀ ∈ = = =

∀ ∈∀ ∈ == == ==

∀ ∈ =  = =

 

ℕ ℕ ℕ

.

ℕ

ا ت 7- ا d Fأ :

x

lim x ln 1 1 x

→+∞→+∞

 

++ ++

 

 

و

2

x 0

lim 1 ln x x

+ + +

→ +

→→

→

 

 

 

++ ++

 

 

 

 

 

 

و

(((( ))))

³

x 0

lim x ln x

+ + +

→ +

→→

→

و

(((( ))))

³

x

lim ln x x

→+∞→+∞

.

(((( ))))

((((

²

))))

xlim x ln x

→+∞

→+∞→+∞

→+∞ −−−−

و

x

lim x ln x

x 1

→+∞

 

 

+ ++

 + 

 

و

2 3 x 0

lim x ln x

+ ++

→ +

→

→→

و

(((( ))))

²³

x

lim ln x x

→+∞

→+∞→+∞

.

→+∞

د 5 ا ا +, 5

(((( ))))

un n 1

≥≥≥

= D 4+5 ا

≥ n

:

n 1 u ln

n + + +

 + 

 

 

===

=  

 

 

 

.

1 ( أ d Fأ

u1 2

و .

u

ب d Fأ

n 1 n

u ₊₊₊₊ −−−−u

D ر i - !ا و

(((( ))))

un n 1

≥

≥≥

.

≥

2 ( XY

n 1 2 3 n

:

s ====u ++++u ++++u ++++... u++++

d Fأ

sn

D i - !ا .c

n

nlim sn

→+∞→+∞

→+∞

.

→+∞

   

  

/ + 02

/ هر,



(6)

ˆbn ÿa ˆbn ÿa Z

Z @ @

Ùi@óïäbrÜa@óå Üa Ùi@óïäbrÜa@óå Üa

@ @

--20201111 201

2010

@@ @@ @@ @@

0

óibïä

ârïéÜaæia@óïÝïèdnÜa@óîíäbrÜa

@ãíÝÈ@bîŠíÜb

ﺌ ŒïÐ

óï bî H‘î†cI



א

א

אא

IV (

La dérivée logarithmique d’une fonction

.

ط @

/0

ق 8 V D S اد

u

ل 1 ا 6 $

،

I

6 $ م 5- و ا

ل 1 .

I

- : ق 8 V D S

u

ل 1 ا 6 $

نذإ ،

I

6 $ A اد

u

.

I

نأ D و 6 $ م 5-

u

ن\4 ،

I

Pا jL-D kL &

u

ل 1 ا 6 $ ةر .

I

XY

(((( )))) (((( (((( )))) )))) :

f x ====ln u x

.

l آ اذإ ل 1 ا 6 $ 5HS ,

u

ن\4

I

:

-

(((( )))) (((( )))) :

x I , f x ln u x

∀ ∈ =

∀ ∈∀ ∈ ==

∀ ∈ =

. نأ D و V D S

u

6 $ ق 8 و

I

(((( )))) ]]]] [[[[

u I ⊂⊂⊂⊂ 0 ,+∞+∞+∞+∞

،

و ل 1 ا 6 $ ق 8 V D S

ln

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- و

(((( )))) (((( (((( )))) )))) (((( )))) (((( )))) :

(((( ))))

u x x I , f x ln u x u x

u x

′ ′ ′ ′′′′

′ ′ ′

∀ ∈ = × =

∀ ∈∀ ∈ == ×× ==

∀ ∈ = × =

l آ اذإ ل 1 ا 6 $ 5HS , !

u

ن\4

I

:

-

(((( )))) (((( (((( )))) )))) :

x I , f x ln u x

∀ ∈ = −

. نأ D و 6 $ ق 8 V D S

u

و

I

(((( )))) ]]]] [[[[

u I 0 ,

− ⊂ +∞

−− ⊂⊂ +∞+∞

− ⊂ +∞

و ل 1 ا 6 $ ق 8 V D S

ln

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- و

(((( )))) (((( (((( )))) )))) (((( )))) (((( )))) :

(((( ))))

u x

x I , f x ln u x u x

u x

′ ′ ′ ′′′′

′ ′ ′

∀ ∈ = − − × =

∀ ∈∀ ∈ = −= − −− ×× ==

∀ ∈ = − − × =

l آ اذإ ق 8 V D S اد

u

ل 1 6 $

،

I

6 $ م 5- و ا

ل 1 ا ا ن\4 ،

I

(((( ))))

x ֏֏֏֏ ln u x

ق 8 V D S 6 $

6 $ 8 9 ا 7 اد و

I

ا ا =ه

I

(((( )))) :

(((( ))))

u x

x u x

֏ ′′′′

֏֏

.

֏

د F

(((( ))))

f ′′′′ x

E0 ل 1 ا /

x

& ا =4

I

ا ت :

1 (

(((( )))) ((((

²

))))

f x ====ln x ++++x++++1

و

]]]] [[[[

I = −∞ +∞= −∞ +∞= −∞ +∞= −∞ +∞,

.

2 (

(((( ))))

f x ====ln 1−−−−ln x

و

]]]] [[[[

I ==== e ,+∞+∞+∞+∞

.

/0 ل 1 6 $ ق 8 V D S اد

u

،

I

6 $ م 5- و ا

ل 1 .

I

ا ا

(((( )))) :

(((( ))))

u x

x u x

֏ ′′′′

֏

ا ر ا 8 9 ا 6

֏

ل 1 ا 6 $

u I

/0 6 $ ق 8 V D S اد

u

ل 1 6 $ م 5- و ،

I

ا ل 1 .

I

لاو ا ا 2"ا

(((( )))) :

(((( ))))

u x

x u x

֏ ′′′′

֏֏

ل 1 ا 6 $

֏

لاو ا =ه

I

(((( )))) :

(((( ))))

x ֏֏֏֏ ln u x ++++c

((((

^c^∈^∈^∈^∈^ℝ^ℝ^ℝ^ℝ

)))) Z F

و ا ا 2"ا لا

x 1

2 x ++++1

֏

ل 1 ا 6 $

֏ , 1

2

 

−∞ −−∞ −−∞ −

−∞ − 

 

 

لاو ا =ه

1

:

x ln 2 x 1 c

2 ++++ ++++

֏

֏֏

،

֏

((((

^c^∈^∈^∈^∈^ℝ^ℝ^ℝ^ℝ

))))

ا 2"ا لاو ا

x ֏֏֏֏tan x

ل 1 ا 6 $

2 2, π ππ π π ππ π

 

−

−−

− 

 

 

لاو ا =ه

x ֏֏֏֏ −−−−ln cos x ++++c

((((

^c^∈^∈^∈^∈^ℝ^ℝ^ℝ^ℝ

)))) ، .



óïámŠbÈíÜÜaßaì‡Üa N óïámŠbÈíÜÜaßaì‡Üa

@ @

   



             

(7)

د 5 ا ا ا +, 5 4+5 ا

f

ل 1 ا 6 $

]]]]

^{2 ,}^+∞^+∞^+∞^+∞

[[[[

= D

(((( )))) :

x2

f x ====4 x

−−

.

1 ( 8 8& ا دا $"ا د F و

a

و

b

Z &D

c

]]]] [[[[ (((( ))))

^b ^c

:

x 2 , , f x a

2 x 2 x

∀ ∈ +∞ = + +

− +

2 ( ا 2"ا لاو ا i - !ا ل 1 ا 6 $

f

]]]]

^{2 ,}^+∞^+∞^+∞^+∞

[[[[

.

ا 2"ا لاو ا د F

$

f

ل 1 ا 6 ا ت & ا =4

I

:

1 (

(((( ))))

3 4

f x x

x 1

==

== ++ ++

و

]]]] [[[[

I = −∞ +∞= −∞ +∞= −∞ +∞= −∞ +∞,

.

2 (

(((( ))))

¹

f x ==== x ln x

و

]]]] [[[[

I ==== 0 ,1

/0 =8 8& ا + f د 5 ا ا ا

f

= D 4+5 ا x

(((( )))) (((( )))) :

f x ====x ++++ln ln x

.

1 ( ا ا > +5 O F نأ l,cأ ه

f

]]]] [[[[

Df ==== 1 ,+∞+∞+∞+∞

.

2 ( d Fأ

(((( ))))

f ′′′′ x

تا+ f i - !ا .c ا ا

.

f

3 ( ا ا 6-&- g 7 V ا عو+L ا سردأ .

f

4 ( ا ا 6-&- h9 أ .C- 5 . 5 =4

f

/ د 5 ا / ا ا +, 5 و

u

=8 8& ا + f

f

= D / 4+5 ا

x

:

(((( ))))

²

u x ====x −−−−2 x ++++3

و

(((( )))) ((((

²

))))

f x ====ln x −−−−2 x ++++3

.

1 ( أ نأ / D

(((( )))) :

x ,u x 0

∀ ∈ >

∀ ∈ℝℝℝℝ >

نأ i - !ا و

]]]] [[[[ :

Df = −∞ +∞= −∞ +∞= −∞ +∞= −∞ +∞,

ب d Fأ

(((( )))) :

u 1++++ 2

(((( )))) و

u 1−−−− 2

.

2 ( / 7- ا d Fأ

(((( )))) :

xlim f x

→+∞→+∞

(((( )))) و

xlim f x

→−∞→−∞→−∞

.

→−∞

3 ( d Fأ

) x ( f′′′′

ا ا تا+ f سردأ .c .

f

4 ( د 5 ا وذ . 8 ا نأ / D

"

x ====1

"

6-&- Ec ر &

ا ا .

f

5 ( أ l,cأ

ا يو ا

(((( ))))

²

:

*

2 3

ln x ln 1

f ( x ) x x

x , 2

x x x

 

− +

 

 

∀ ∈ = +

∀ ∈ℝℝℝℝ = +

.

ب ا ا 6-&- / g 7 V ا / $+L ا سردأ .

f

6 ( أ ا ا 6-&- نأ / D ,8

f

7- Eآ =cا Fإ د F .c ف H5 ا = H8 E .

ب ا ا 6-&- .!رأ .C- 5 . 5 =4

f

.

7 ( /0 ا ا ر AS

g

ل 1 ا 6 $

f

[[[[ [[[[

I ==== 1 ,+∞+∞+∞+∞

.

أ ا ا نأ / D ل 1 ا / ED 8

g

ل 1 &

I

q & .

J

.

ب + ,5 د F

(((( ))))

g⁻⁻⁻⁻1 x

D E0

x

/

x

.

J

) r<

:

ln 2 ≈≈≈≈0.7

و

ln 3 ≈≈≈≈1.1

(

/ + 03

/ + 04

/ + 05

/ +

06

(8)

ˆbn ÿa ˆbn ÿa Z

Z @ @

Ùi@óïäbrÜa@óå Üa Ùi@óïäbrÜa@óå Üa

@ @

--20201111 201

2010

@@ @@ @@ @@

0

óibïä

ârïéÜaæia@óïÝïèdnÜa@óîíäbrÜa

@ãíÝÈ@bîŠíÜb

ﺌ ŒïÐ

óï bî H‘î†cI



א

א

אא óïámŠbÈíÜÜaßaì‡Üa N

óïámŠbÈíÜÜaßaì‡Üa

@ @

V ( )

a

a >>>>0

و

a ≠≠≠≠1

( :

a

1 (

/0 د 5 L 3 و 5HS , 8 8F اد $

a

1

س !: . ر ا اد

O + D 7 O + = ا د 5 ا ا ا =ه

a Loga

ا و 6 $ 4+5

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

= D

(((( )))) :

a

Log x ln x

= ln a

=

2 (

(((( ))))

Loga 1 ====0

(((( ))))

Loga a ====1

(((( ))))

a

Log e 1

=ln a

==

=

(((( ))))

e

x 0 , Log x ln x ln x

∀ > = ln e =

∀ > = =

- و

e

G

ln====Log

.

E0 و

x

ل 1 ا /

y

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

E0 و ، /

r

ℚℚℚ

- ،

ℚ

:

1 (((( )))) (((( )))) (((( )))) (

a a a

Log xy ====Log x ++++Log y

2

a a

(

Log 1 Log x

x

 

= −

= −= −

 = −

 

 

3 (((( )))) (((( )))) (

a a a

Log x Log x Log y

y

 

= −

== −−

= −

 

 

4 (

(((( ))))

^r

a a

Log x ====rLog x

= b D :

(((( ))))

2 2

A Log 1 Log 10

5

 

= +

=  +

 

(((( ))))

⁵

1 3

B ====Log 3

3 a (

ا ا

Loga

ل 1 ا 6 $ ق 8 V D S

]]]]

^{0 ,}^+∞^+∞^+∞^+∞

[[[[

- و

((((

a

)))) (((( )))) :

ln x 1

x 0 , Log x

ln a x ln a

 ′′′′

 

∀ > ′′′′ = =

∀ > = =

∀ > =  =

 

/ ا / و 1 ا i- G- و :

F

a>>>>1

+∞

+∞+∞

+∞

a

1

0

x

+++

+ +++

+

((((

Loga

))))

′′′′

(((( ))))

x

+∞

+∞+∞

+∞

1

0

−∞−∞

(((( ))))

Loga x

F

0 <<<<a <<<<1

+∞

1

a

0

x

++ ++

((((

Loga

))))

′′′′

(((( ))))

x

+∞

0

1

−∞−∞

(((( ))))

Loga x

 

   



      

^.

 

Fonction logarithme de base



ACË ÕæKPA«ñÊË@ éË@X é@PX