Résumé Mathématique sur les fonctions exponentielles 1
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Course
Mathématique
Institution
High School
Ceci est un cours complet sur les fonctions exponentielles niveau lycée il contient plusieurs notions très importantes qui pourront soit vous aidez à comprendre la leçon ou pour renforcer sa compréhension.
Fonctions logarithmes
La fonction logarithme
Df 0, , continue et dérivable sur Df 0, avec ln x '
1
népérienne f x ln x x
ln1 0 , lne 1 avec e 2, 718... e est un nombre irrationnel .
a 0 et b 0 et r 1 a
Signe de ln(x) lnab lna lnb , ln ln a , ln ln a ln b , lnar r lna
a b
x 0 1
a,b 0, ,ln a = ln b a b
ln x 0
a,b 0, ,a b ln a ln b
f x ln u x , x Df x Du et u x
u' x u' x
f ' x ln u x sont F x ln u x c
'
donc primitives de
u x u x
lim ln x lim ln x lim x ln x 0
x x 0 x0
ln x ln x lim xn ln x 0
lim 0 lim n 0 x 0
x x x x
ln x ln x 1
lim 1 lim 1
x 1 x1 x0 x
Logarithmes de base a
Logarithmes de base a : ln x ln x
a 0,1 1, r f x loga x , a 0,1 1, ; loge x ln x
lna ln e
a 10 donc log10 x Log x ( logarithme décimale )
loga x y loga x loga y et loga xr = r loga x
x
log a 1 log a y et loga loga x loga y
y y
f x ax La fonction exponentielle népérienne f x e
x
Les fonctions La fonction réciproque de x ln x est la fonction x ex définie de
exponentielles de base a
0, .donc x , ex 0 .
est : f (x) ax exlna
a 0,1 1, r
Df , continue et dérivable sur Df avec e ' e
x x
f x e
u x
x, y , x Df x Du
u' x eu x donc primitives de u' x eu x sont F x eu x c
f ' x e
u x
y '
ax ay ax y ; a x axy
1 ax e y
x
x ln y
x , ln ex x x 0, , elnx x
x x y
x a ; y a
a a x y0
0 a 1 , x, y
ea
1
ex
r
a a xy
x y
eab ea eb e b , e ab
= erx , ex ex e2x
eb eb
a 1 , x, y
lim ex 0 lim ex
ex ex enx
n
ax ay x y x x
ex ex
lim x ex 0 lim x e 0 ; n
n x *
a ' ln a a
x x
x x
n
xlna ex ex ex 1
Rq : f (x) a e
x
lim lim ; n *
lim 1
x x x xn x0 x
-1-
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