Ad tili
a
∈
]
0
;
+
∞
[
{\displaystyle a\in ]0;+\infty [}
x
↦
1
/
x
{\displaystyle x\mapsto 1/x}
a
{\displaystyle a}
Tasɣnt talugaritm tga unrar lli illan ɣ izddar n tasɣnt
x
↦
1
/
x
{\displaystyle x\mapsto 1/x}
a
{\displaystyle a}
ln
a
=
∫
1
a
1
x
d
x
.
{\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}
Iɣ darnɣ
a
<
1
{\displaystyle a<1}
Tṭṭaf tasɣnt talugaritmiyt aydatn kullu tn n tsɣnin tilugaritmin .[ 1]
ln
(
a
b
)
=
ln
a
+
ln
b
.
{\textstyle \ln(ab)=\ln a+\ln b.}
Nzḍar ad nml ayad s tibḍit n uɣrd lli f nsawl ɣ umzwaru s snat tfulin ɣ akud ann nɣrd s usnfl n umutti
x
=
t
a
{\displaystyle x=ta}
ln
(
a
b
)
=
∫
1
a
b
1
x
d
x
=
∫
1
a
1
x
d
x
+
∫
a
a
b
1
x
d
x
=
∫
1
a
1
x
d
x
+
∫
1
b
1
a
t
d
(
a
t
)
=
∫
1
a
1
x
d
x
+
∫
1
b
1
t
d
t
=
ln
a
+
ln
b
.
{\displaystyle {\begin{aligned}\ln(ab)=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}\,d(at)\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}}
Tasɣnt n ulugaritm agaman
ssnfl
Zɣ aylli izrin ar nttafa mas d tasɣnt
x
↦
l
n
(
x
)
{\displaystyle x\mapsto ln(x)}
]
0
;
+
∞
[
{\displaystyle ]0;+\infty [}
∀
x
∈
R
+
∗
,
ln
′
(
x
)
=
1
x
{\displaystyle \forall x\in \mathbb {R} _{+}^{*},\ \ln '(x)={\frac {1}{x}}}
S ɣikAd tilia
l
n
{\displaystyle ln}
]
0
;
+
∞
[
{\displaystyle ]0;+\infty [}
l
n
{\displaystyle ln}
]
0
;
+
∞
[
{\displaystyle ]0;+\infty [}
ssnfl
Ad tg
f
{\displaystyle f}
f
(
x
)
=
ln
(
a
x
)
{\displaystyle \ f(x)=\ln(ax)}
a
{\displaystyle a}
x
{\displaystyle x}
f
(
x
)
=
ln
(
x
)
+
k
/
k
∈
R
{\displaystyle \ f(x)=\ln(x)+k\ /\ k\in \ R}
Acku
f
(
1
)
=
k
{\textstyle f(1)=k}
l
n
(
a
)
=
k
{\displaystyle ln(a)=k}
∀
(
a
;
b
)
∈
]
0
:
+
∞
[
2
,
ln
(
a
b
)
=
ln
(
a
)
+
ln
(
b
)
{\displaystyle \forall (a;b)\in ]0:+\infty [^{2},\ \ln(ab)=\ln(a)+\ln(b)}
Zɣ uyda yad rad naf aydatn ddaw as:
∀
(
a
;
b
)
∈
]
0
:
+
∞
[
2
,
ln
(
a
b
)
=
ln
(
a
)
−
ln
(
b
)
{\displaystyle \forall (a;b)\in ]0:+\infty [^{2},\ \ln \left({\frac {a}{b}}\right)=\ln(a)-\ln(b)}
Iɣ iga
n
{\displaystyle n}
∀
a
∈
]
0
;
+
∞
[
,
∀
n
∈
Z
,
ln
(
a
n
)
=
n
ln
(
a
)
{\displaystyle \forall a\in ]0;+\infty [,\ \forall n\in \mathbb {Z} ,\ \ln(a^{n})=n\ln(a)}
∀
a
∈
]
0
;
+
∞
[
,
∀
r
∈
Q
,
ln
(
a
r
)
=
r
ln
(
a
)
{\displaystyle \forall a\in ]0;+\infty [,\ \forall r\in \mathbb {Q} ,\ \ln(a^{r})=r\ln(a)}
Iɣ iga
n
{\displaystyle n}
∀
a
∈
R
∗
,
∀
n
∈
Z
∗
,
ln
(
a
n
)
=
2
n
ln
|
a
|
{\displaystyle \forall a\in \mathbb {R} ^{*},\ \forall \;n\in \mathbb {Z} ^{*},\ \ln(a^{n})=2n\ln |a|}
∀
(
a
1
,
a
2
,
.
.
.
.
,
a
k
)
∈
]
0
:
+
∞
[
k
,
∑
n
=
1
k
ln
(
a
n
)
=
ln
(
∏
n
=
1
k
a
n
)
{\displaystyle \forall (a_{1},a_{2},....,a_{k})\in ]0:+\infty [^{k},\ \sum _{n=1}^{k}\ln(a_{n})=\ln(\prod _{n=1}^{k}a_{n})}
∏
{\displaystyle \prod }
ln
1
=
0
{\displaystyle \ln 1=0}
ln
e
=
1
{\displaystyle \ln e=1}
ln
(
x
y
)
=
ln
x
+
ln
y
i
x
>
0
d
y
>
0
{\displaystyle \ln(xy)=\ln x+\ln y\quad {\text{i }}\;x>0\;{\text{d }}\;y>0}
ln
(
x
y
)
=
y
ln
x
i
x
>
0
{\displaystyle \ln(x^{y})=y\ln x\quad {\text{i }}\;x>0}
ln
x
<
ln
y
i
0
<
x
<
y
{\displaystyle \ln x<\ln y\quad {\text{i }}\;0<x<y}
lim
x
→
0
ln
(
1
+
x
)
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}
lim
α
→
0
x
α
−
1
α
=
ln
x
i
x
>
0
{\displaystyle \lim _{\alpha \to 0}{\frac {x^{\alpha }-1}{\alpha }}=\ln x\quad {\text{i }}\;x>0}
x
−
1
x
≤
ln
x
≤
x
−
1
i
x
>
0
{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{i}}\quad x>0}
ln
(
1
+
x
α
)
≤
α
x
i
x
≥
0
d
α
≥
1
{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{i}}\quad x\geq 0\;{\text{d }}\;\alpha \geq 1}
Agaman : naturel
Tasɣnt, Tawwuri : fonction
Afaris : produit
Timrnit : somme
Azadur : base
Tamnzut : fonction primitive
Azilal : intervalle
Amusnak : mathematicien
Taflwit : tableau
Tazmilt : notion
Asnml : definition
Anrar : aire
Ayda : propriete
Aɣrd : integrale
Tafult : partie
Amutti : variable
Taɣlalt : continuite
Uzdir : negatif
Tsbk : monotonie
Zllm : deriver
Igmmn : croissante
Umnig : positif
Tamhlt : opperation
Amḍan : nombre
Amatay : general
Waxiḍ : impair
Ana : paire
![{\displaystyle ]0;+\infty [}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2b9afa8192cc3f8173fadddad0d1f5c499d881)
![{\displaystyle a\in ]0;+\infty [}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b33d73dc2873c676cad660baddaeaaa446355d1)
![{\displaystyle {\begin{aligned}\ln(ab)=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}\,d(at)\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2b53dd1976a99950b5c8217d287c4af90d381d)
.svg)
![{\displaystyle \forall (a;b)\in ]0:+\infty [^{2},\ \ln(ab)=\ln(a)+\ln(b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b82a9f70dc4d37520a622aa11d284072b8940537)
![{\displaystyle \forall (a;b)\in ]0:+\infty [^{2},\ \ln \left({\frac {a}{b}}\right)=\ln(a)-\ln(b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be7a5b58797e75ef324c761d5000fddeb7651f6e)
![{\displaystyle \forall a\in ]0;+\infty [,\ \forall n\in \mathbb {Z} ,\ \ln(a^{n})=n\ln(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6f04d2e3f902b8050e7c7103f83607f21f2c1b)
![{\displaystyle \forall a\in ]0;+\infty [,\ \forall r\in \mathbb {Q} ,\ \ln(a^{r})=r\ln(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38cef3f3f5057e91f503c68cbb07f8afc8a63d8d)
![{\displaystyle \forall (a_{1},a_{2},....,a_{k})\in ]0:+\infty [^{k},\ \sum _{n=1}^{k}\ln(a_{n})=\ln(\prod _{n=1}^{k}a_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b986b1d2f8d3f8be67a75962fc87a625f45ce7ac)
