a
c
=
b
⇔
c
=
l
o
g
a
b
;
a
>
0
;
a
≠
1
;
b
>
0
{\displaystyle a^{c}=b\Leftrightarrow c=log_{a}b;a>0;a\neq 1;b>0}
a
l
o
g
a
b
=
b
l
o
g
a
1
=
0
l
o
g
a
a
=
1
{\displaystyle {\begin{matrix}a^{log_{a^{b}}}=b\\log_{a}1=0\\log_{a}a=1\end{matrix}}}
l
o
g
b
c
=
l
o
g
a
c
l
o
g
a
b
{\displaystyle log_{b}c={\frac {log_{a}c}{log_{a}b}}}
l
o
g
a
b
n
=
n
l
o
g
a
b
{\displaystyle {\begin{matrix}log_{a}{b^{n}}=nlog_{a}b\end{matrix}}}
l
o
g
a
(
b
c
)
=
l
o
g
a
b
+
l
o
g
a
c
{\displaystyle {\begin{matrix}log_{a}(bc)=log_{a}b+log_{a}c\end{matrix}}}
l
o
g
a
b
c
=
l
o
g
a
b
−
l
o
g
a
c
{\displaystyle log_{a}{\frac {b}{c}}=log_{a}b-log_{a}c}
Arvuti pahavara "Dx2" dubleerib ennast 7 sekundi tagant. Mitme minuti pärast on pahavara "Dx2" ennast üle miljoni suutnud paljundada, kui neid on arvutis juba 2 tükki? Vastus anda kümnendkohta täpsusega.
2
x
>
1000000
{\displaystyle 2^{x}>1000000}
a
b
=
c
⇒
b
=
l
o
g
a
c
⇒
b
=
l
o
g
10
c
l
o
g
10
a
{\displaystyle a^{b}=c\Rightarrow b=log_{a}c\Rightarrow b={\frac {log_{10}c}{log_{10}a}}}
x
>
l
o
g
10
1000000
l
o
g
10
2
⇒
x
>
6
0
,
30102999566
⇒
x
>
19
,
931568569324
{\displaystyle x>{\frac {log_{10}1000000}{log_{10}2}}\Rightarrow x>{\frac {6}{0,30102999566}}\Rightarrow x>19,931568569324}
x * 7 sekundit = aeg
aeg = 19,931568569324 * 7 sekundit =~ 139,5 sekundit = 2,3 minutit
Vastus: Pahavara on ennast suutnud paljundada 2,3 minuti jooksul üle miljoni.
1)
3
x
=
1
81
{\displaystyle 3^{x}={\frac {1}{81}}}
2)
10
x
=
81
{\displaystyle 10^{x}=81}
3)
log
x
27
=
3
{\displaystyle \log _{x}27=3}
4)
ln
x
=
−
2
ln
3
{\displaystyle \ln x=-2\ln 3}
5)
2
x
=
5
⋅
3
x
{\displaystyle 2^{x}=5\cdot 3^{x}}
