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

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(1)

--20201111 201

2010

0

x ֏֏֏֏ln u x

x u' x u x

֏

֏

֏

֏

a

(2)

## N

I

x 1 x

֏

֏֏

֏

0 ,+∞+∞+∞+∞

0 ,+∞+∞+∞+∞

1

x 1 x

֏

֏

֏

֏

0 ,+∞+∞+∞+∞

1

ln

ln

0 ,+∞+∞+∞+∞

ln 1 ====0

ln

0 ,+∞+∞+∞+∞

1

x 0 , , ln x x

∀ ∈ +∞ ′′′′ =

∀ ∈∀ ∈ +∞+∞ ==

∀ ∈ +∞ =

ln

0 ,+∞+∞+∞+∞

### ]]]] [[[[ :

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

a

b

0 ,+∞+∞+∞+∞

### (((( )))) :

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

x 0 , F x ln kx

∀ > =

∀ > =

∀ > =

∀ > =

x 0 ,u x kx

∀ > =

∀ > =

∀ > =

∀ > =

k∈∈∈∈ℝℝℝℝ++++

x 0 , F x ln u x

∀ > =

∀ >∀ > ==

∀ > =

F

0 ,+∞+∞+∞+∞

### (((( )))) (((( (((( )))) )))) (((( ))))

1

x 0 , F x ln u x u x

′ ′ ′ x

′ ′ ′

′ ′ ′

′ ′ ′

∀ > = × =

∀ >∀ > == ×× ==

∀ > = × =

c / F x ln x c

∃ ∈ = +

∃ ∈ = +

∃ ∈ = +

∃ ∈ℝℝℝℝ = +

x ====1

ln k ====c

### (((( ))))

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

x ====a

k ====b

### (((( )))) :

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 @@

ârïéÜaæia@óïÝïèdnÜa@óîíäbrÜaóibïä

@ãíÝÈ@bîŠíÜb

ŒïÐ

óï bî H‘î†cI

א

א

f : x x

ln x

֏

֏֏

֏

### (((( ))))

g : x ֏֏֏֏ x −−−−1 ln x

h : x ֏֏֏֏ ln ln x

ℝℝ ℝℝ

2

### ))))

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

### (((( )))) (((( ))))

ln x −−−−1 −−−−ln 3 x++++1 ====0

2

x 1

ln 0

x 1

+ ++

 + 

 

 

 

=

==

 =

 

 

 

+ + +

 + 

 

 

 

ℝ ℝ ℝ

### (((( ))))

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

### (((( )))) (((( ))))

ln x −−−−1 <<<<ln 2 x−−−−1

2

x 1

ln 0

x 1

−−

 − 

 

 

 

 ≥

 

 

 

++ ++

 

 

 

 

II

(3)

i n i n

i i

i 1 i 1

ln a ln a

= =

== ==

= =

= =

= =

= =

= =

 

 

 

=

=

=

 =

 

 

 

 

a1 2

a

n

a

a

b

0 ,+∞+∞+∞+∞

r ℚ ℚ ℚ

1

ln ln a

a

 

 

 

 = −= −= −= −

  

  

 

 

 

 

a

ln ln a ln b b

 

 

 

 ==== −−−−

  

  

 

 

 

 

r

ln a ====r ln a

1

ln a ln a

= 2

==

=

1 1

### :

a 0 ,0 ln 1 ln a ln a ln

a a

 

 

 

 

∀ > = = × = +

∀ > = = × = +

∀ > = = × = +

∀ > = =  × = +

 

 

 

 

ln 1 ln a a

 

 

 

 = −= −= −= −

 

 

 

  

 

 

 

a 1 1

### :

ln ln a ln a ln ln a ln b

b b b

   

   

   

   

= × = + = −

= × = + = −

= × = + = −

= × = + = −

   

   

   

   

   

   

   

   

a

b

+ + +

+

ℝ ℝℝ

r ====n∈∈∈∈ℕℕℕℕ

1 2 n

### :

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

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 = − = − =

∀ > = = − = − =

∀ > = = − = − =

∀ > = = − = − =

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

∀ > =

∀ > =

∀ > =

∀ > =

xy >>>>0

### (((( )))) :

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

ln x ln x ln y y

 

 

 

 

= −

= −

= −

= −

 

 

 

 

 

 

 

 

ln x2 ====2 ln x

x

0 ,+∞+∞+∞+∞

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

ℝ ℝ ℝ

### (((( )))) :

ln − +− +− +− +x 3 ====2

### (((( )))) :

ln − +− +− +− +x 3 ≥≥≥≥2

ln

0 ,+∞+∞+∞+∞

### د-ا ت.

xlim ln x

→+∞

→+∞→+∞

→+∞ = +∞= +∞= +∞= +∞

x 0

lim ln x

+ + +

+

= −∞= −∞= −∞= −∞

x

lim ln x 0 x

→+∞

→+∞→+∞

→+∞ ====

x

### :

x 0 x 0

lim ln x lim ln1 lim ln x

+ + x

+ +

+ +

+ + →+∞→+∞→+∞→+∞

= − = − = −∞

= −= − = −= − = −∞= −∞

= − = − = −∞

(4)

--20201111 201

2010

0

א

א

ln

ln

0 ,+∞+∞+∞+∞

1

x 0 , ln x 0

x

∀ > ′′′′ = >

∀ >∀ > == >>

∀ > = >

+∞

+∞+∞

+∞

1

0

x

+ ++

+

+ + +

+

ln ′′′′ x

+∞

+∞+∞

+∞

0

−∞

−∞−∞

−∞

ln x

ln

x 0

lim ln x

++ ++

= −∞= −∞

= −∞= −∞

### ا ا 6-&- يد \$ بر 8 d ار"ا ر & نذإ .

ln

xlim ln x

→+∞→+∞

→+∞→+∞ = +∞= +∞= +∞= +∞

x

lim ln x 0 x

→+∞→+∞→+∞

→+∞ ====

ln

+∞+∞

+∞+∞

e

ln

0 ,+∞+∞+∞+∞

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

x 0 x

ln 0 , lim ln x , lim ln x ,

+ + +

+ →+∞→+∞→+∞→+∞

 

 

 

 

+∞ = = −∞ +∞

+∞ = = −∞ +∞

+∞ = = −∞ +∞

+∞ = = −∞ +∞

### ]]]] [[[[

1∈ −∞ +∞∈ −∞ +∞∈ −∞ +∞∈ −∞ +∞,

ln x ====1

0 ,+∞+∞+∞+∞

e

ln e ====1

e∉∉∉∉ℚℚℚℚ

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 ≤ ⇔ ≤

ln

ln

1

y ====x−−−−1

ln

e 1

y x

=e

=

==

O y

1 1

1

y=lnx

e

y ====x1

y 1x

=e

==

=

(5)

f

### (((( )))) :

f x ====x−−−−ln x

Df

Df

f

f

f

x 0

lim x ln x 0

+ + +

+

====

x 1

lim ln x 1 x 1

====

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

x

lim x ln 1 1 x

→+∞→+∞

→+∞→+∞

 

 

 

 

++ ++

 

 

 

 

 

 

 

 

2

x 0

lim 1 ln x x

+ + +

+

 

 

 

++ ++

 

 

 

 

 

 

3

x 0

lim x ln x

+ + +

+

3

x

lim ln x x

→+∞→+∞

→+∞→+∞

2

xlim x ln x

→+∞

→+∞→+∞

→+∞ −−−−

x

lim x ln x

x 1

→+∞

→+∞

→+∞

→+∞

 

 

 

 

 

 

 

 

+ ++

 + 

 

 

 

2 3 x 0

lim x ln x

+ ++

+

23

x

lim ln x x

→+∞

→+∞→+∞

→+∞

un n 1

n

n 1 u ln

n + + +

 + 

 

 

===

=  

 

 

 

u1 2

u

n 1 n

u ++++ −−−−u

un n 1

n 1 2 3 n

### :

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

sn

n

nlim sn

→+∞→+∞

→+∞

→+∞

(6)

--20201111 201

2010

0

א

א

### IV(

La dérivée logarithmique d’une fonction

u

I

I

u

I

u

I

u

I

u

I

f x ====ln u x

u

I

x I , f x ln u x

∀ ∈ =

∀ ∈∀ ∈ ==

∀ ∈ =

u

I

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

u I ⊂⊂⊂⊂ 0 ,+∞+∞+∞+∞

ln

0 ,+∞+∞+∞+∞

### (((( ))))

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

u x

′ ′ ′ ′′′′

′ ′ ′

′ ′ ′

′ ′ ′

∀ ∈ = × =

∀ ∈∀ ∈ == ×× ==

∀ ∈ = × =

u

I

x I , f x ln u x

∀ ∈ = −

∀ ∈ = −

∀ ∈ = −

∀ ∈ = −

u

I

u I 0 ,

− ⊂ +∞

−− ⊂⊂ +∞+∞

− ⊂ +∞

ln

0 ,+∞+∞+∞+∞

### (((( ))))

u x

x I , f x ln u x u x

u x

′ ′ ′ ′′′′

′ ′ ′

′ ′ ′

′ ′ ′

∀ ∈ = − − × =

∀ ∈∀ ∈ = −= − −− ×× ==

∀ ∈ = − − × =

u

I

I

x ֏֏֏֏ ln u x

I

I

u x

x u x

֏ ′′′′

֏֏

֏

f ′′′′ x

x

I

2

### ))))

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

### ]]]] [[[[

I = −∞ +∞= −∞ +∞= −∞ +∞= −∞ +∞,

### (((( ))))

f x ====ln 1−−−−ln x

### ]]]] [[[[

I ==== e ,+∞+∞+∞+∞

u

I

I

u x

x u x

֏ ′′′′

֏

֏

֏

u I

u

I

I

u x

x u x

֏ ′′′′

֏֏

֏

I

### (((( ))))

x ֏֏֏֏ ln u x ++++c

c

x 1

2 x ++++1

֏

֏

֏

֏ , 1

2

 

 

 

 

−∞ −−∞ −−∞ −

−∞ − 

 

 

 

 

 

 

 

1

x ln 2 x 1 c

2 ++++ ++++

֏

֏֏

֏

c

x ֏֏֏֏tan x

### ل 1 ا 6 \$

2 2, π ππ π π ππ π

 

 

 

 

−−

− 

 

 

 

 

 

 

 

### لاو ا =ه

x ֏֏֏֏ −−−−ln cos x ++++c

c

(7)

f

2 ,+∞+∞+∞+∞

x2

f x ====4 x

−−

−−

a

b

c

b c

x 2 , , f x a

2 x 2 x

∀ ∈ +∞ = + +

∀ ∈ +∞ = + +

∀ ∈ +∞ = + +

∀ ∈ +∞ = + +

− +

− +

− +

− +

f

2 ,+∞+∞+∞+∞

f

I

3 4

f x x

x 1

==

== ++ ++

### ]]]] [[[[

I = −∞ +∞= −∞ +∞= −∞ +∞= −∞ +∞,

1

f x ==== x ln x

I ==== 0 ,1

f

### (((( )))) (((( )))) :

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

f

### ]]]] [[[[

Df ==== 1 ,+∞+∞+∞+∞

f ′′′′ x

f

f

f

u

f

x

### (((( ))))

2

u x ====x −−−−2 x ++++3

2

### ))))

f x ====ln x −−−−2 x ++++3

x ,u x 0

∀ ∈ >

∀ ∈ >

∀ ∈ >

∀ ∈ℝℝℝℝ >

### ]]]] [[[[ :

Df = −∞ +∞= −∞ +∞= −∞ +∞= −∞ +∞,

u 1++++ 2

u 1−−−− 2

xlim f x

→+∞→+∞

→+∞→+∞

xlim f x

→−∞→−∞→−∞

→−∞

) x ( f′′′′

f

x ====1

f

2

*

2 3

ln x ln 1

f ( x ) x x

x , 2

x x x

 

 

 

 

− +

− +

− +

− +

 

 

 

 

 

 

 

 

∀ ∈ = +

∀ ∈ = +

∀ ∈ = +

∀ ∈ℝℝℝℝ = +

f

f

f

g

f

### [[[[ [[[[

I ==== 1 ,+∞+∞+∞+∞

g

I

J

g1 x

x

x

J

ln 2 ≈≈≈≈0.7

ln 3 ≈≈≈≈1.1

(8)

--20201111 201

2010

0

א

א

a

a >>>>0

a ≠≠≠≠1

a

a

1

a Loga

0 ,+∞+∞+∞+∞

a

Log x ln x

= ln a

=

=

=

Loga 1 ====0

Loga a ====1

a

Log e 1

=ln a

==

=

### (((( ))))

e

x 0 , Log x ln x ln x

∀ > = ln e =

∀ > = =

∀ > = =

∀ > = =

e

ln====Log

x

y

0 ,+∞+∞+∞+∞

r

ℚℚℚ

### (((( )))) (((( )))) (((( )))) (

a a a

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

a a

Log 1 Log x

x

 

 

 

 

= −

= −= −

 = −

 

 

 

 

 

 

 

### (((( )))) (((( )))) (

a a a

Log x Log x Log y

y

 

 

 

 

= −

== −−

= −

 

 

 

 

 

 

 

 

r

a a

Log x ====rLog x

2 2

A Log 1 Log 10

5

 

 

 

 

= +

= +

= +

=  +

 

 

 

 

5

1 3

B ====Log 3

Loga

0 ,+∞+∞+∞+∞

a

ln x 1

x 0 , Log x

ln a x ln a

 ′′′′

 

 

 

∀ > ′′′′ = =

∀ > = =

∀ > = =

∀ > =  =

 

 

 

 

a>>>>1

+∞

+∞+∞

+∞

a

1

0

x

+++

+ +++

+ +++

+

Loga

′′′′

x

+∞

+∞+∞

+∞

1

0

−∞−∞

−∞−∞

Loga x

0 <<<<a <<<<1

+∞

+∞

+∞

+∞

1

a

0

x

++ ++

++ ++

++ ++

Loga

′′′′

x

+∞

+∞

+∞

+∞

0

1

−∞−∞

−∞−∞

Loga x

.

###  

Fonction logarithme de base

### ACË ÕæKPA«ñÊË@ éË@X é@PX

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