Www.science-ki.blogspot.com MATH SM a (9)

7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)

1/9

- 2Bac PC+SVT: Fonctions logarithmiques : 10:

Lyce My RachidLyce My RachidLyce My RachidLyce My Rachid Ague Ague Ague Aguelmous Mohamedlmous Mohamedlmous Mohamedlmous Mohamed _ iaalou iaalou iaalou iaalou @ yahoo.fr yahoo.fr yahoo.fr yahoo.fr- 1 -

(((( )))) x ln u x

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

u' x x

u x

.a

.

( )====x10 a .

.

a

a


2/9



I(- Fonction logarithme nprien

1 x 0]]]] [[[[ , ++++ 0]]]] [[[[ , ++++

.1 x 1

x0]]]] [[[[ , ++++ .1

.ln:

0]]]] [[[[ ln , ++++.ln 1 0====

0]]]] [[[[ ln , ]]]] [[[[: ++++ (((( )))) 1

x 0 , , ln x x + = + = + = + =

0]]]] [[[[ ln , a ]]]] [[[[:.++++ ,b 0 , , a b ln a ln b + < = = > = =

c / F x ln x c (((( )))) = + = + = + = + x 1==== ln k c====. ln(((( )))) kx ln x ln k= += += += +.

x a==== k b====:(((( ))))ln : ab ln a ln b= += += += +.


3/9



(((( )))):

i n i n

i ii 1 i 1

ln a ln a= == == == =

= == == == =

====

1 a 2 a ... n a .

a b0]]]] [[[[ , : r ++++

1(-1ln ln a a

= = = =

2(- aln ln a ln b b

= = = =

3(-(((( )))) rln a r ln a==== 4(-1ln a ln a 2

====

1(-:1 1 a 0 ,0 ln 1 ln a ln a ln

a a

> = = = + > = = = + > = = = + > = = = +

1ln ln a

a

= = = =

.

2(-: a 1 1ln ln a ln a ln ln a ln b b b b

= = + = = = + = = = + = = = + =

a b++++ .

-)3 =r n= = = = :1 2 n a a .... a a = = == = = == = = == = = nln : = a n ln a====

r n = = = = : r n n n1

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

> = = = = > = = = = > = = = = > = = = =

p r q

====: r qr p a ,q ln a ln a ln a p ln a > = = = > = = = > = = = > = = = r a 0 ,ln a r ln a > = > = > = > =

xy 0>>>>:(((( ))))ln xy ln x ln y= += += += + xln ln x ln y

y

= = = =

2ln x 2 ln x====

x0]]]] [[[[ , n : n ++++ 1ln x ln x n

====.

=A ln 2 2 ln 2 2 : + += + += + += + +.

(((( )))) (((( )))) 2009 2009

B ln 2 1 ln 2 1= + += + += + += + + ln(((( )))): x 3 = + = + = + = + 2 ln(((( )))): x 3 2 + + + +

II(-

0]]]] [[[[ ln , ++++.-)1

xlim ln x

++++ + =+ =+ =+ =

x 0lim ln x

++++ = = = =

x

ln xlim 0

x++++====

:

x x 0 x 0

1lim ln x lim ln lim ln x x+ ++ ++ ++ + ++++

= = = = = = = = = = = = .


4/9



-)2 ln ln ]]]] [[[[0 , (((( )))) (((( )))): ++++ 1 x 0 , ln x 0

x

> = > > = > > = > > = >

++++ 1 0 x ++++ ++++ (((( )))) (((( ))))ln x

++++ 0

ln x

-)3 ln :

x 0lim ln x

++++ = = = = .ln

xlim ln x++++ = += += += + xln x

lim 0 x++++ ==== ln ++++ 4(- e

ln 0]]]] [[[[ , ++++

(((( ))))]]]] [[[[: ]]]] [[[[ x x 0

ln 0 , lim ln x , lim ln x ,++++ ++++

+ = = ++ = = ++ = = ++ = = + . , 1]]]] [[[[ + + + + ln x ====1 0]]]] [[[[ , ++++ e.

ln: e e ====1 e 2.718.(((( )))) k x 0 , k , ln e k > = > = > = > = k x 0 , k , ln x k x e > = = > = = > = = > = = .

k x 0 , k , ln x k x e > > > > 5(- ln

. = = = =y x 1 : 1 ln

y x 1: e ln e

====.

O

y

1

11

y=lnx

e

y x 1= = = =

1 y x

e=== =


5/9



01 = = = =f x x ln x (((( )))):

-)1 f

D f

D. -)2 f -)3 . -)4

III(-

x 0lim x ln x 0

++++ ====

x 1

ln xlim 1

x 1====

(((( ))))

x 0

ln 1 xlim 1

x++++

====

n

x 0

n , lim x ln x 0++++

= = = = n x

ln x n , lim 0

x

++++

= = = =

t x 0 x 0 x 0

1ln

1 ln t xlim x ln x lim x ln lim lim

1 x t x

+ + ++ + ++ + ++ + + ++++

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

.

(((( )))) x 1

ln xlim ln 1 1

x 1= == == == =

.

(((( )))) (((( )))) x 0

ln 1 xlim 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+ + ++ + ++ + ++ + +

= = = = = = = = = = = = .

n

n n x x t

ln x 1 ln x 1 ln t n , lim lim lim 0

x n x n t

+ + ++ + ++ + ++ + +

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

.

:

x

1lim x ln 1

x++++

++++ x 0 2

1lim ln x

x++++

++++ 3 (((( ))))

x 0lim x ln x

++++(((( ))))

3

x

ln xlim

x++++.

(((( ))))(((( )))) 2 xlim x ln x++++ x x

lim x ln x 1++++ ++++

2 3

x 0lim x ln x

++++(((( ))))

2 3

x

ln xlim

x++++.

02

n (((( )))) n 1u n : n 1

u ln n

++++ ====

.

++++n 1 nu u -. 1u 2u - -)1 n (((( )))) n 1u .

2(-: n 1 2 3 n s u u u ... u= + + + += + + + += + + + += + + + + .++++n nlim s n s n


6/9



III(- .La drive logarithmique dune fonction

u I . I :u I u . I u I

u I .:(((( )))) (((( ))))(((( )))) x ln u x====.

: I u :(((( )))) (((( )))) x I , f x ln u x = = = = (((( )))) I u . ]]]] [[[[u I 0 , + + + +

ln ]]]] [[[[0 , (((( ))))(((( )))) (((( )))): ++++ (((( )))) (((( ))))(((( ))))u x

x I , f x ln u x u xu x

= = = = = = = =

: I u:(((( )))) (((( ))))(((( )))) x I , f x ln u x = = = = u . I (((( )))) ]]]] [[[[u I 0 , + + + +

ln ]]]] [[[[0 , (((( ))))(((( )))) (((( )))): ++++ (((( )))) (((( ))))(((( ))))u x

x I , f x ln u x u xu x

= = = = = = = =

u I I (((( ))))(((( )))) x ln u x

I I (((( )))): (((( ))))

u x x

u x

.

: I x (((( )))) 1(- (((( )))) (((( )))) 2 f x ln x x 1= + += + += + += + + ]]]] [[[[ I ,= += += += +.2(- (((( )))) x ln 1 ln x= = = = ]]]] [[[[ I e ,= += += += +.

u I . I

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

u x x

u x

I u

u I . I

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

u x x

u x

I ln(((( ))))(((( )))): u x c++++ (((( )))) c

x 1

2 x 1++++, 1

2

x ln 2 1: x 1 c 2

+ ++ ++ ++ + (((( )))) c

2 2 x tan x x ln cos x c

+ + + + (((( )))) c .


7/9



03

2 ]]]] [[[[ , ++++ :(((( )))) 2 x

x 4 x

====

.

-)1 ]]]] [[[[:a b c (((( )))) b c x 2 , , f x a 2 x 2 x

+ = + + + = + + + = + + + = + + + + + +

-)2 2 ]]]] [[[[ , ++++. 04

I : 1(- (((( ))))

3

4

x x

x 1====

++++.+ =+ =+ =+ =, I ]]]] [[[[

2(- (((( )))) 1 x x ln x

====I 0 ,1 ]]]] [[[[ ====

05 .+ =+ =+ =+ =x x ln ln x (((( )))) (((( )))): x

-)1 f D 1 ]]]] [[[[ ,= += += += +.x (((( ))))-)2 .-)3 . f -)4

06

(((( )))) : x u 2u x x 2 x 3= += += += + (((( )))) (((( )))) 2 x ln x 2 x 3= += += += +. --)1 x ,u x 0 > > > > (((( )))): + =+ =+ =+ =, f D ]]]] [[[[:

.u 1 2(((( )))) ++++u 1 2(((( )))): - (((( )))): -)2

xlim f x

++++(((( ))))

xlim f x

.

x(f( -)3 . -)4 "====1" .

--)5 (((( )))): 2

*

2 3ln 1ln x f ( x ) x x

x , 2 x x x

+ + + +

= + = + = + = + .

- . --)6 .

- . 7(- g I 1 ]]]] ]]]] f ,= += += += +.

- g I . .g x x x 1(((( ))))-

):ln 2 0.7 ln 3 1.1(


8/9



V (- a) a 0>>>> a 1(: aFonction logarithme de base1(-

a 1 a 0]]]] [[[[ a Log , ++++

:(((( )))) aln x

Log xln a

====

2(- (((( )))) a Log 1 0==== (((( )))) a Log a 1==== (((( )))) a

1 Log e

ln a====

(((( ))))eln x

x 0 ,Log x ln xln e

> = = > = = > = = > = eln = Log====.

x y0]]]] [[[[ , : r ++++

1(-(((( )))) (((( )))) (((( )))) a a a Log xy Log x Log y= += += += + 2(- a a1

Log Log x x

= = = =

3(-(((( )))) (((( )))) a a a x

Log Log x Log y y

= = = =

4(- (((( )))) r a a Log x rLog x====

:

(((( )))) 2 21

A Log Log 10 5

= += += += + (((( ))))

51

3 B Log 3====

-)3 a

a Log ]]]] [[[[0 , a (((( )))) (((( )))): ++++ln x 1

x 0 , Log xln a x ln a

> = = > = = > = = > = =

: a 1>>>>

++++ a 1 0 x ++++ ++++ ++++ (((( )))) (((( ))))

a Log x

++++ 1 0

(((( )))) a Log x

0 a 1<


9/9



-)4 a Log a 1>>>> 0 a 1< >

==== = = = =

.

--)1 f D 0 ]]]] ]]]] xlim f (((( )))) + =+ =+ =+ =, x++++ . - .

- -)2 .

- ]]]] [[[[: (((( )))) (((( ))))(((( )))) 2

2 1 ln x x 0 , , f x

x ln x

+ = + = + = + =

.

- . - -)4 1 C(((( )))) 1

2

.

f C ).ln (((( )))) - 2 0.7 ,e 2.7 .(

O 1

1(((( )))) 2 y Log x=== =

O 1

1

(((( ))))1 2

y Log x=== =

Documents

Www.science-ki.blogspot.com MATH SM a (9)