# Exponential and logarithmic formula

[EN/CZ]

## Exponential formulas

Formula Function domain
$a^r . a^s = a^{r + s}$ $a \in \R ; r,s \in \N$
$(a^r)^s = a^{rs}$ $a \in \R ; r,s \in \N$
$\frac{a^r}{a^s} = a^{r - s}$ $a \in \R - \{0\}; r,s \in \N ; r > s$
$(a . b)^r = a^r . b^r$ $a,b \in \R ; r,s \in \N$
$(\frac{a}{b})^r = \frac{a^r}{b^r}$ $a \in \R; b \in \R - \{0\} ; r,s \in \N$
$a^0 = 1$ $a \in \R - \{0\}$
$0^m = 0$ $m \in \N$
$a^{-n} = \frac{1}{a^n}$ $a \in \R - \{0\}; n \in \N$
$(\frac{a}{b})^{-m} = (\frac{b}{a})^m$ $a,b \in \R - \{0\}; m \in \Z$
$1^m$ $m \in \R$
$\sqrt[2n]{a^{2n}} = ∣a∣$ $a \in \R; n \in \N$
$(\sqrt{a})^r = \sqrt{a^r}$ $a \ge 0 ; r,s \in \R$

## Logarithmic formulas

Formula Function domain
$\log_{a}a = 1$ $a \in \R^{+} - \{1\}$
$\log_{a}1 = 0$ $a \in \R^{+} - \{1\}$
$\log_{a}a^r = r$ $a \in \R^{+} - \{1\} ;r \in \R$
$\log_{a}xy = \log_{a}x + \log_{a}y$ $a \in \R^{+} - \{1\} ;x,y \in \R^{+}$
$\log_{a}\frac{x}{y} = \log_{a}x - \log_{a}y$ $a \in \R^{+} - \{1\} ;x,y \in \R^{+}$
$\log_{a}x = \frac{log_{b}x}{\log_{b}a}$ $a,b \in \R^{+} - \{1\} ;x \in \R^{+}$
$\log_{a}b = \frac{1}{\log_{b}a}$ $a,b \in \R^{+} - \{1\}$