Sivu:Lukion taulukot.pdf/23

22 MERKINTÖJÄ, MÄÄRITELMIÄ JA KAAVOJA

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Logaritmi

$k^{x}=a\Leftrightarrow x=\log _{k}a{\text{ eli }}k^{\log _{k}a}=a\ (k>0,k\neq 1,a>0)$

$\log _{k}k^{x}=x$ $k$ on kantaluku

$\log _{10}a=\lg a;\log _{2}a={\text{lb}}\ a;\log _{e}a=\ln a$
 $\ln a$ on luonnollinen logaritmi

1. $\log ab=\log a+\log b$ (kantaluku jätetty merkitsemättä)
2. $\log {\frac {a}{b}}=\log a-\log b$
3. $\log a^{n}=n\log a$
4. $\log _{k}a={\frac {\lg a}{\lg k}}={\frac {\ln a}{\ln k}}$

Determinantit

Kaksirivinen

${\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}=a_{1}b_{2}-a_{2}b_{1}$

Kolmirivinen

${\begin{aligned}{\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}&=a_{1}{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}+a_{2}{\begin{vmatrix}b_{3}&b_{1}\\c_{3}&c_{1}\end{vmatrix}}+a_{3}{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\\&=a_{1}{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}-a_{2}{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}+a_{3}{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\end{aligned}}$

Summia ja sarjoja

1.	${\begin{aligned}S_{n}&=a+(a+d)+(a+2d)+\ldots +\underbrace {(a+(n-1)d)} _{u}\\&=n\cdot {\frac {a+u}{2}}\end{aligned}}$	aritmeettinen summa $a$ on ensimmäinen ja $u$ viimeinen termi
2.	${\begin{aligned}S_{n}&=a+aq+aq^{2}+\ldots +aq^{n-1}\\&={\frac {a(1-q^{n})}{1-q}}\ (q\neq 1)\end{aligned}}$	geometrinen summa
3.	$a+aq+aq^{2}+\ldots$ $s={\frac {a}{1-q}}$	geometrinen sarja; suppenee, kun $|q|<1$; summa $s$
4.	$1+2+3+\ldots +n={\frac {n(n+1)}{2}}$
5.	$1^{2}+2^{2}+3^{2}+\ldots +n^{2}={\frac {n(n+1)(2n+1)}{6}}$