www.science-ki.blogspot.com math sm a (9)
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
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- 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
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
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- 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- 2 -
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= += += += +.
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
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- 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- 3 -
(((( )))):
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+ ++ ++ ++ + ++++
= = = = = = = = = = = = .
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
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- 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- 4 -
-)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=== =
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
5/9
- 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- 5 -
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
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
6/9
- 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- 6 -
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 .
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
7/9
- 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- 7 -
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(
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
8/9
- 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- 8 -
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<
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7/23/2019 Www.science-ki.blogspot.com MATH SM a (9)
9/9
- 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- 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=== =