Naturālā logaritma funkcijas grafiks. Naturālais logaritms ir logaritms, kura bāze ir skaitlis
e
{\displaystyle \ e}
e
=
2
,
7182818...
{\displaystyle e=2,7182818...\,}
x
{\displaystyle \ x}
ln
x
{\displaystyle \ \ln x}
log
e
x
=
ln
x
{\displaystyle \log _{e}x=\ln x\,}
ln
x
{\displaystyle \ \ln x}
e , iegūtu skaitli x .
Visiem reāliem x
ln
e
x
=
x
.
{\displaystyle \ln e^{x}=x.\,\!}
un visiem pozitīviem x
e
ln
x
=
x
.
{\displaystyle e^{\ln x}=x.\,}
ln
x
{\displaystyle \ \ln x}
y
=
1
x
{\displaystyle y={\frac {1}{x}}}
a
{\displaystyle a}
ln
a
=
∫
1
a
1
x
d
x
.
{\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}
ln
x
{\displaystyle x}
ln
x
{\displaystyle x}
ln
(
x
y
)
=
ln
x
+
ln
y
{\displaystyle \ln(xy)=\ln x+\ln y\!\,}
Visiem
x
>
0
{\displaystyle x>0}
ln
x
≤
x
−
1
{\displaystyle \ln x\leq x-1}
ln
′
x
=
1
x
{\displaystyle \ln 'x={\frac {1}{x}}}
lim
x
→
0
ln
(
1
+
x
)
x
=
1.
{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1.\,}
Ja
a
>
0
{\displaystyle a>0}
lim
x
→
∞
ln
x
x
a
=
0.
{\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x^{a}}}=0.\,}
ln
(
1
+
x
)
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
x
n
=
x
−
x
2
2
+
x
3
3
−
⋯
j
a
|
x
|
≤
1
{\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots \quad {\rm {ja}}\quad \left|x\right|\leq 1\quad }
izņemot
x
=
−
1
{\displaystyle x=-1}
Dažreiz naturālie logaritmi tiek izmantoti lai atvasinātu kādu funkciju. Piemēram,
(
x
x
)
′
=
(
e
x
ln
x
)
′
=
e
x
ln
x
⋅
(
x
ln
x
)
′
=
x
x
⋅
(
1
+
ln
x
)
{\displaystyle (x^{x})'=(e^{x\ln x})'=e^{x\ln x}\cdot (x\ln x)'=x^{x}\cdot (1+\ln x)}
