. ]0 , + ∞ [ ﻤﻌرﻓﺔ و ﻤﺘﺼﻠﺔ و ﻗﺎﺒﻠﺔ ﻟﻼﺸﺘﻘﺎق و ﺘزاﻴدﻴﺔ ﻗطﻌﺎً ﻋﻠﻰ ّ ln اﻟداﻟﺔ x ﻤﻌرﻓﺔ و ﻤﺘﺼﻠﺔ و ﻗﺎﺒﻠﺔ ﻟﻼﺸﺘﻘﺎق و ﺘزاﻴدﻴﺔ ﻗطﻌﺎً ﻋﻠﻰ ّ e اﻟداﻟﺔ اﻷﺴﻴﺔ
lim e x = +∞
6
lim e x = 0+
y=x
lim ln( x) = −∞
x →0 +
lim x ⋅ e = 0 x→− ∞ n∈ : ex lim n = +∞ x→+ ∞ x ex −1 =1 lim x→ 0 x n
x
x∈ , y∈ ex ⋅ e y = e( x+ y ) ex = e( x− y ) y e 1 e( − x ) = x e x r * ∀r ∈ : (e ) =e( r⋅x ) 1 ( ⋅x ) 2 1 ( ⋅x ) n
a اﻟداﻟﺔ اﻷﺴﻴﺔ ﻟﻸﺴﺎس a > 0 , a ≠1 ∀x ∈ : a x =e( x⋅ln a ) = a 0 1= ; a1 a a x =t ⇔ x =log a (t ) 10 x =t ⇔ x =log(t ) (= a x ) / (ln a ) ⋅ a x
3
lnx
2 1
-2
-1
0
2
1
3
4
5
6
-1 -2 @
ln x = a ⇔ x = e a (ln x ) e= x , x>0
ln(= ex ) x , x ∈
x>0, y>0 ln( x × y )= ln( x) + ln( y ) x y − ln( ) ln( ) = ln( x) − ln( y ) = y x 1 ln( ) = − ln( x) x * ∀r ∈ : ln( x r ) = r ⋅ ln( x) 1 ln( x )= ⋅ ln( x) 2 1 ln( n x )= ⋅ ln( x) n
÷ÄÄÔÄÄ–ÌÄnÄ‹bi
x →− ∞
÷ÄÄÔÄÄ–ÌÄnÄ‹bi
4
ex lim = +∞ x→+ ∞ x lim x ⋅ e x = 0−
ex = e
x→+∞
5
x→− ∞
n
lim ln( x) = +∞
exp
x→+ ∞
ex = e
x e x :] ﻫﻲ اﻟداﻟﺔ اﻷﺴﻴﺔ0, + ∞ [ ﻋﻠﻰln اﻟداﻟﺔ اﻟﻌﻛﺴﻴﺔ ﻟﻠداﻟﺔ
ln( x) = 0+ x lim x ⋅ ln( x) = 0− lim
x→+∞
x →0 +
lim x n ⋅ ln( x) = 0− x →0 + n ∈ * : ln( x) = 0+ xlim n →+∞ x ln( x) ln(1 + x) lim 1= ; lim 1 = x→1 x − 1 x →0 x
= ln(e) 1= ; ln(1) 0 e1 = e ≈ 2,7182 ; e0 = 1 ln( x) > 0 ⇔ x > 1 x ∀x ∈ : e > 0 ln( x) = 0 ⇔ x = 1 x x > 0 ⇔ e >1 ln( x) < 0 ⇔ 0 < x < 1 ln( x) > 1 ⇔ x > e x =0 ⇔ e x =1 x ln( x) =1 ⇔ x =e x < 0 ⇔ 0 < e <1 ln( x) < 1 ⇔ 0 < x < e a∈ ; b∈ a>0 ; b>0 e a = eb ⇔ a = b ln( a= ) ln(b) ⇔ = a b e a < eb ⇔ a < b ln( a ) < ln(b) ⇔ a < b
a اﻟﻠوﻏﺎرﺘم ﻟﻸﺴﺎس 1 ي اﻟﻌﺸر اﻟﻠوﻏﺎرﺘم x > 0 : (ln x) = a > 0 , a ≠1 x ln( x) ln( x) ∀x > 0 : log( x) = f / ( x) / ∀x > 0 : log a ( x) = ln(10) f ( x) > 0 : ( ln (f ( x)) ) = ln( a ) f ( x)= log(1) 0= ; log(10) 1 log a (1) 0= ; log a (a ) 1 = u / ( x) / log(0,1) 1 = − u ( x) ≠ 0 : ( ln (| u ( x) |) ) = log a (a r ) = r u ( x) r log(10 ) = r x / x log a ( x) = b ⇔ x = a b (e ) = e 1 1 (log( x)) /= ⋅ 1 1 u( x) / u( x) / (e = ) u ( x) ⋅ e x ln10 (log a ( x)) /= ⋅ x ln a /
c ∈ ﺤﻴثx ln(| u ( x) |) + c : ﻫﻲ اﻟدواﻝI ﻋﻠﻰx
u / ( x) : اﻟدواﻝ اﻷﺼﻠﻴﺔ ﻟﻠداﻟﺔ. وﻻ ﺘﻨﻌدم ﻋﻠﻴﻪI داﻟﺔ ﻗﺎﺒﻠﺔ ﻟﻼﺸﺘﻘﺎق ﻋﻠﻰ ﻤﺠﺎﻝu u ( x)
