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U matematici postoje nekoliko logaritamskih jednačina .
Ljudi koriste logaritme da bi uprostili račun. Na primer, dva broja mogu biti pomnožena samo koristeći tablicu logaritama i sabiranje.
log
b
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x
y
)
=
log
b
(
x
)
+
log
b
(
y
)
{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)\!\,}
zbog
b
m
⋅
b
n
=
b
m
+
n
{\displaystyle b^{m}\cdot b^{n}=b^{m+n}}
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)}
zbog
b
m
b
n
=
b
m
−
n
{\displaystyle {\begin{matrix}{\frac {b^{m}}{b^{n}}}\end{matrix}}=b^{m-n}}
log
b
(
x
y
)
=
y
log
b
(
x
)
{\displaystyle \log _{b}(x^{y})=y\log _{b}(x)\!\,}
zbog
(
b
n
)
y
=
b
n
y
{\displaystyle (b^{n})^{y}=b^{ny}\!\,}
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}}}
zbog
x
y
=
x
1
/
y
{\displaystyle {\sqrt[{y}]{x}}=x^{1/y}}
Logaritmi i eksponenti (antilogaritmi) sa istom osnovom se poništavaju.
b
log
b
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x
)
=
x
{\displaystyle b^{\log _{b}(x)}=x}
zbog
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\!\,}
zbog
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}}
Ova jednačina se koristi za izračunavanje logaritama na elektronskim kalkulatorima. Na primer, većina kalkulatora ima dugmad za ln i za log10 , ali ne i za log2 . Da bismo našli log2 (3), treba izračunati log10 (3) / log10 (2) (ili ln(3)/ln(2), što je zapravo ista stvar).
Iz ove formule proizilazi nekoliko stvari:
log
a
b
=
1
log
b
a
{\displaystyle \log _{a}b={\frac {1}{\log _{b}a}}}
log
a
n
b
=
1
n
log
a
b
{\displaystyle \log _{a^{n}}b={\frac {1}{n}}\log _{a}b}
a
log
b
c
=
c
log
b
a
{\displaystyle a^{\log _{b}c}=c^{\log _{b}a}}
log
b
(
1
)
=
0
{\displaystyle \log _{b}(1)=0\!\,}
zbog
b
0
=
1
{\displaystyle b^{0}=1\!\,}
log
b
(
b
)
=
1
{\displaystyle \log _{b}(b)=1\!\,}
zbog
b
1
=
b
{\displaystyle b^{1}=b\!\,}
lim
x
→
0
+
log
a
x
=
−
∞
if
a
>
1
{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty \quad {\mbox{if }}a>1}
lim
x
→
0
+
log
a
x
=
∞
if
a
<
1
{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=\infty \quad {\mbox{if }}a<1}
lim
x
→
∞
log
a
x
=
∞
if
a
>
1
{\displaystyle \lim _{x\to \infty }\log _{a}x=\infty \quad {\mbox{if }}a>1}
lim
x
→
∞
log
a
x
=
−
∞
if
a
<
1
{\displaystyle \lim _{x\to \infty }\log _{a}x=-\infty \quad {\mbox{if }}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}
Poslednji limes se često shvata kao „logaritam raste sporije od bilo kog stepena ili korena 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}
∫
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}
što se za а=е svodi na
∫
ln
x
d
x
=
x
(
ln
x
−
1
)
+
C
{\displaystyle \int \ln x\,dx=x(\ln x-1)+C}
x
1
+
x
≤
log
(
1
+
x
)
≤
x
за све
−
1
<
x
{\displaystyle {\frac {x}{1+x}}\leq \log(1+x)\leq x{\mbox{ за све }}-1<x}
2
x
2
+
x
≤
x
1
+
x
+
x
2
/
12
≤
log
(
1
+
x
)
≤
x
1
+
x
≤
x
2
2
+
x
1
+
x
за
0
≤
x
, обрнуто за
−
1
<
x
≤
0
{\displaystyle {\frac {2x}{2+x}}\leq {\frac {x}{\sqrt {1+x+x^{2}/12}}}\leq \log(1+x)\leq {\frac {x}{\sqrt {1+x}}}\leq {\frac {x}{2}}{\frac {2+x}{1+x}}{\mbox{ за }}0\leq x{\mbox{, обрнуто за }}-1<x\leq 0}
Obe nejednakosti su prilično oštre oko x=0, ali ne i za veliko x.[ 1] [ 2]
![{\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)
