KiÕn thøc mò vµ logarit
A.kh¸i niÖm:
*)a∈R,n∈N *)a∈R/{ }0 ,n∈N
a0 = 1 ≠ 0 ; 1 = n n
a
a− = 1
n
a =
*)a> 0 ;m,n∈N
m
n
a
a =
m n m n
a
a− = 1
*)
>
≠
>
=
⇔
= 0
1 , 0 log
M
a a
M a x
a
*)
≅
=
= 71828 , 2
ln log
lg log10
e
M M
M M
B.TÝnh chÊt:
1.a m.a n =a m+n
2 m n
n
m
a a
a = −
3.( )a n m =a mn
4.( )n n m
b a b
a =
5 n n
n
b
a b
a =
1 a M M
log 7.loga(M.N)= loga M + loga N
2.aloga M =M 8 M N
N
M
a a
3.loga1 = 0 9.loga Mα = α loga M
4.loga a= 1 10.loga M 1loga M
α
5
b
M M
a
a b
log
log log = 11.loga M = loga b logb M
6
a
b
b a
log
1 log = 12.alogb c =clogb a
c.§¹o hµm:
( )a x ,=a x lna ( )e x , =e x ( )
a x
x
a
ln
1 ' log = ( )
x
x ' 1
ln =
D.Ph¬ng tr×nh:
1.§a vÒ cïng c¬ sè:
≠
>
=
1
;
0 a
a
a
a f x g x
⇔ f( ) ( )x = g x
2.LÊy log hai vÕ:
( )
>
=
⇔
≠
>
=
0
log 1
;
x g x
f a
a
x
g
3.§Æt Èn phô: ( )( ) ( ) ( )
=
=
⇔
=
0
0
t f
a t a
f
x g x
g
1.§a vÒ cïng c¬ sè:
( ) ( )
=
>
>
⇔
≠
>
=
x g x f
x g x
f a
a
x g x
1
; 0
log log
2.Mò hãa hai vÕ:
( ) ( ) ( ) g( )x a
a x f a
x g x f
=
⇔
≠
<
= 1 0
log
=
=
⇔
=
0
log 0
log
t f
x t
x
a
E.BÊt ph¬ng tr×nh:(0 <a≠ 1)
( ) ≥a ( ) ⇔ (a− 1 )[f( ) ( )x −g x ]≥ 0
a f x g x
( ) ( ) ( −( ) ) ( ) ( ) [ −( ) ]≥
>
>
⇔
≥
0 1
0
; 0 log
log
x g x f a
x g x f x
g x
a