Els logaritmes s'utilitzen normalment per simplificar les operacions. Per exemple, els logaritmes ens permeten resoldre un càlcul que inclou multiplicacions simplement amb sumes.
log
b
(
x
y
)
=
log
b
(
x
)
+
log
b
(
y
)
{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)\!\,}
donat que
b
x
⋅
b
y
=
b
x
+
y
{\displaystyle b^{x}\cdot b^{y}=b^{x+y}\!\,}
log
b
(
x
y
)
=
log
b
(
x
)
−
log
b
(
y
)
{\displaystyle \log _{b}\!\left({\begin{matrix}{\frac {x}{y}}\end{matrix}}\right)=\log _{b}(x)-\log _{b}(y)}
donat que
b
x
b
y
=
b
x
−
y
{\displaystyle {\begin{matrix}{\frac {b^{x}}{b^{y}}}\end{matrix}}=b^{x-y}}
log
b
(
x
y
)
=
y
log
b
(
x
)
{\displaystyle \log _{b}(x^{y})=y\log _{b}(x)\!\,}
donat que
(
b
x
)
y
=
b
x
y
{\displaystyle (b^{x})^{y}=b^{xy}\!\,}
log
b
(
x
y
)
=
log
b
(
x
)
y
{\displaystyle \log _{b}\!\left(\!{\sqrt[{y}]{x}}\right)={\begin{matrix}{\frac {\log _{b}(x)}{y}}\end{matrix}}}
donat que
x
y
=
x
1
/
y
{\displaystyle {\sqrt[{y}]{x}}=x^{1/y}}
x
log
b
(
y
)
=
y
log
b
(
x
)
{\displaystyle x^{\log _{b}(y)}=y^{\log _{b}(x)}\!\,}
donat que
x
log
b
(
y
)
=
b
log
b
(
x
)
log
b
(
y
)
=
b
log
b
(
y
)
log
b
(
x
)
=
y
log
b
(
x
)
{\displaystyle x^{\log _{b}(y)}=b^{\log _{b}(x)\log _{b}(y)}=b^{\log _{b}(y)\log _{b}(x)}=y^{\log _{b}(x)}\!\,}
On
b
{\displaystyle b}
x
{\displaystyle x}
y
{\displaystyle y}
b
≠
1
{\displaystyle b\neq 1}
Les següents sumes/restes són especialment útils en teoria de probabilitats quan es tracta d'una suma/resta de probabilitats logarítmiques:
log
b
(
a
+
c
)
=
log
b
a
+
log
b
(
1
+
b
log
b
c
−
log
b
a
)
{\displaystyle \log _{b}(a+c)=\log _{b}a+\log _{b}(1+b^{\log _{b}c-\log _{b}a})}
log
b
(
a
−
c
)
=
log
b
a
+
log
b
(
1
−
b
log
b
c
−
log
b
a
)
{\displaystyle \log _{b}(a-c)=\log _{b}a+\log _{b}(1-b^{\log _{b}c-\log _{b}a})}
En particular:
log
b
(
a
+
c
)
=
log
b
a
+
log
b
(
1
+
c
a
)
{\displaystyle \log _{b}(a+c)=\log _{b}a+\log _{b}\left(1+{\frac {c}{a}}\right)}
log
b
(
a
−
c
)
=
log
b
a
+
log
b
(
1
−
c
a
)
{\displaystyle \log _{b}(a-c)=\log _{b}a+\log _{b}\left(1-{\frac {c}{a}}\right)}
log
b
(
1
)
=
0
{\displaystyle \log _{b}(1)=0\!\,}
donat que
b
0
=
1
{\displaystyle b^{0}=1\!\,}
log
b
(
b
)
=
1
{\displaystyle \log _{b}(b)=1\!\,}
donat que
b
1
=
b
{\displaystyle b^{1}=b\!\,}
Fixem-nos que
log
b
(
0
)
{\displaystyle \log _{b}(0)\!\,}
x
{\displaystyle x\!\,}
b
x
=
0
{\displaystyle b^{x}=0\!\,}
asímptota vertical al gràfic de la funció
f
(
x
)
=
log
b
(
x
)
{\displaystyle f(x)=\log _{b}(x)\!\,}
x
=
0
{\displaystyle x=0\!\,}
Els logaritmes i exponencials (antilogaritmes) amb la mateixa base es cancel·len. Això és degut al fet que els logaritmes i els exponencials són operacions inverses (tal com passa amb la multiplicació i la divisió).
b
log
b
(
x
)
=
x
{\displaystyle b^{\log _{b}(x)}=x}
donat que
a
n
t
i
l
o
g
b
(
log
b
(
x
)
)
=
x
{\displaystyle \mathrm {antilog} _{b}(\log _{b}(x))=x\!\,}
log
b
(
b
x
)
=
x
{\displaystyle \log _{b}(b^{x})=x\!\,}
donat que
log
b
(
a
n
t
i
l
o
g
b
(
x
)
)
=
x
{\displaystyle \log _{b}(\mathrm {antilog} _{b}(x))=x\!\,}
log
a
b
=
log
c
b
log
c
a
{\displaystyle \log _{a}b={\log _{c}b \over \log _{c}a}}
Aquesta relació és necessària per trobar els valor d'un logaritme amb una calculadora. Per exemple, la majoria de calculadores tenen els botons ln i log10 , però no log₂. Per trobar log₂(3), hem de calcular log10 (3) / log10 (2) (o ln(3)/ln(2), que té el mateix resultat).
Tenim
y
=
log
a
b
{\displaystyle y=\log _{a}b\,}
I per tant
a
y
=
b
{\displaystyle a^{y}=b\,}
Si agafem
log
c
{\displaystyle \log _{c}\,}
log
c
a
y
=
log
c
b
{\displaystyle \log _{c}a^{y}=\log _{c}b\,}
Simplificant i resolent:
y
log
c
a
=
log
c
b
{\displaystyle y\log _{c}a=\log _{c}b\,}
y
=
log
c
b
log
c
a
{\displaystyle y={\frac {\log _{c}b}{\log _{c}a}}}
Donat que
y
=
log
a
b
{\displaystyle y=\log _{a}b\,}
log
a
b
=
log
c
b
log
c
a
{\displaystyle \log _{a}b={\frac {\log _{c}b}{\log _{c}a}}}
Aquesta fórmula té unes quantes conseqüències:
log
a
b
=
1
log
b
a
{\displaystyle \log _{a}b={\frac {1}{\log _{b}a}}}
log
a
n
b
=
log
a
b
n
{\displaystyle \log _{a^{n}}b={{\log _{a}b} \over n}}
a
log
b
c
=
c
log
b
a
{\displaystyle a^{\log _{b}c}=c^{\log _{b}a}}
−
log
a
b
=
log
a
(
1
b
)
=
log
1
a
b
{\displaystyle -\log _{a}b=\log _{a}\left({1 \over b}\right)=\log _{1 \over a}b}
log
a
1
b
1
⋯
log
a
n
b
n
=
log
a
π
(
1
)
b
1
⋯
log
a
π
(
n
)
b
n
,
{\displaystyle \log _{a_{1}}b_{1}\,\cdots \,\log _{a_{n}}b_{n}=\log _{a_{\pi (1)}}b_{1}\,\cdots \,\log _{a_{\pi (n)}}b_{n},\,}
On
π
{\displaystyle \scriptstyle \pi \,}
permutació de les bases 1, ..., n . Per exemple
log
a
w
⋅
log
b
x
⋅
log
c
y
⋅
log
d
z
=
log
d
w
⋅
log
a
x
⋅
log
b
y
⋅
log
c
z
.
{\displaystyle \log _{a}w\cdot \log _{b}x\cdot \log _{c}y\cdot \log _{d}z=\log _{d}w\cdot \log _{a}x\cdot \log _{b}y\cdot \log _{c}z.\,}
lim
x
→
0
+
log
a
x
=
−
∞
si
a
>
1
{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty \quad {\mbox{si }}a>1}
lim
x
→
0
+
log
a
x
=
∞
si
a
<
1
{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=\infty \quad {\mbox{si }}a<1}
lim
x
→
∞
log
a
x
=
∞
si
a
>
1
{\displaystyle \lim _{x\to \infty }\log _{a}x=\infty \quad {\mbox{si }}a>1}
lim
x
→
∞
log
a
x
=
−
∞
si
a
<
1
{\displaystyle \lim _{x\to \infty }\log _{a}x=-\infty \quad {\mbox{si }}a<1}
lim
x
→
0
+
x
b
log
a
x
=
0
{\displaystyle \lim _{x\to 0^{+}}x^{b}\log _{a}x=0}
lim
x
→
∞
1
x
b
log
a
x
=
0
{\displaystyle \lim _{x\to \infty }{1 \over x^{b}}\log _{a}x=0}
L'últim límit es resumeix dient que els logaritmes creixen més lentament que qualsevol potència o arrel de x.
d
d
x
ln
x
=
1
x
=
1
x
ln
e
,
x
>
0
{\displaystyle {d \over dx}\ln x={1 \over x}={1 \over x\ln e},\qquad x>0}
d
d
x
log
b
x
=
1
x
ln
b
,
b
>
0
,
b
≠
1
{\displaystyle {d \over dx}\log _{b}x={1 \over x\ln b},\qquad b>0,b\neq 1}
ln
x
=
∫
1
x
1
t
d
t
{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}dt}
∫
log
a
x
d
x
=
x
(
log
a
x
−
log
a
e
)
+
C
{\displaystyle \int \log _{a}x\,dx=x(\log _{a}x-\log _{a}e)+C}
Per recordar integrals més grans, és necessari definir:
x
[
n
]
=
x
n
(
log
(
x
)
−
H
n
)
{\displaystyle x^{\left[n\right]}=x^{n}(\log(x)-H_{n})}
On
H
n
{\displaystyle H_{n}}
nombre harmònic . Per exemple:
x
[
0
]
=
log
x
{\displaystyle x^{\left[0\right]}=\log x}
x
[
1
]
=
x
log
(
x
)
−
x
{\displaystyle x^{\left[1\right]}=x\log(x)-x}
x
[
2
]
=
x
2
log
(
x
)
−
3
2
x
2
{\displaystyle x^{\left[2\right]}=x^{2}\log(x)-{\begin{matrix}{\frac {3}{2}}\end{matrix}}\,x^{2}}
x
[
3
]
=
x
3
log
(
x
)
−
11
6
x
3
{\displaystyle x^{\left[3\right]}=x^{3}\log(x)-{\begin{matrix}{\frac {11}{6}}\end{matrix}}\,x^{3}}
Llavors,
d
d
x
x
[
n
]
=
n
x
[
n
−
1
]
{\displaystyle {\frac {d}{dx}}\,x^{\left[n\right]}=n\,x^{\left[n-1\right]}}
∫
x
[
n
]
d
x
=
x
[
n
+
1
]
n
+
1
+
C
{\displaystyle \int x^{\left[n\right]}\,dx={\frac {x^{\left[n+1\right]}}{n+1}}+C}
![{\displaystyle \log _{b}\!\left(\!{\sqrt[{y}]{x}}\right)={\begin{matrix}{\frac {\log _{b}(x)}{y}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa4327beb2984c4f0548bee011606d7588db706)
![{\displaystyle {\sqrt[{y}]{x}}=x^{1/y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ac55954aa0ab68281337cdf011c7e92b309446)
![{\displaystyle x^{\left[n\right]}=x^{n}(\log(x)-H_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be26aeae0c7b88d50e760d2ce40df2af4c44b0bb)
![{\displaystyle x^{\left[0\right]}=\log x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/078fe3653cf35a30aea1b7f03ea554ae7670b967)
![{\displaystyle x^{\left[1\right]}=x\log(x)-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c12b6bda581e741822ed456b8e7c42955525db0)
![{\displaystyle x^{\left[2\right]}=x^{2}\log(x)-{\begin{matrix}{\frac {3}{2}}\end{matrix}}\,x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c21bdba06d346a4ba6099ffecd4612135f157a)
![{\displaystyle x^{\left[3\right]}=x^{3}\log(x)-{\begin{matrix}{\frac {11}{6}}\end{matrix}}\,x^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1dd85612d205f8fab7b2af8590f297eec9c209a)
![{\displaystyle {\frac {d}{dx}}\,x^{\left[n\right]}=n\,x^{\left[n-1\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1466307fb829dca298b9bfcce3e58bd33c52d8a)
![{\displaystyle \int x^{\left[n\right]}\,dx={\frac {x^{\left[n+1\right]}}{n+1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92036b7056a0179b00f4d6739640046d2e8553ba)