Exponenciální a logaritmické vzorce


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Exponenciální vzorce

Vzorec Definiční obor
ar.as=ar+s a^r . a^s = a^{r + s}  aR;r,sNa \in \R ; r,s \in \N
(ar)s=ars(a^r)^s = a^{rs} aR;r,sNa \in \R ; r,s \in \N
aras=ars\frac{a^r}{a^s} = a^{r - s} aR{0};r,sN;r>sa \in \R - \{0\}; r,s \in \N ; r > s
(a.b)r=ar.br(a . b)^r = a^r . b^r a,bR;r,sNa,b \in \R ; r,s \in \N
(ab)r=arbr(\frac{a}{b})^r = \frac{a^r}{b^r} aR;bR{0};r,sNa \in \R; b \in \R - \{0\} ; r,s \in \N
a0=1a^0 = 1 aR{0}a \in \R - \{0\}
0m=00^m = 0 mNm \in \N
an=1ana^{-n} = \frac{1}{a^n} aR{0},nNa \in \R - \{0\}, n \in \N
(ab)m=(ba)m(\frac{a}{b})^{-m} = (\frac{b}{a})^m a,bR{0};mZa,b \in \R - \{0\}; m \in \Z
1m1^m mRm \in \R
a2n2n=a\sqrt[2n]{a^{2n}} = ∣a∣ aR;nNa \in \R; n \in \N
(a)r=ar(\sqrt{a})^r = \sqrt{a^r} a0;r,sRa \ge 0 ; r,s \in \R

Logaritmické vzorce

Vzorec Definiční obor
logaa=1\log_{a}a = 1 aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}
loga1=0\log_{a}1 = 0 aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}
logaar=r\log_{a}a^r = r aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}
logaxy=logax+logay\log_{a}xy = \log_{a}x + \log_{a}y aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}
logaxy=logaxlogay\log_{a}{\frac{x}y} = \log_{a}x - \log_{a}y aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}
logax=logbxlogba\log_{a}x = \frac{log_{b}x}{\log_{b}a} aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}
logab=1logba\log_{a}b = \frac{1}{\log_{b}a} aR+{1};bR+{1}a \in \R^{+} - \{1\} ;b \in \R^{+} - \{1\}