MATHEMATICS Exponential and logarithmic formula February 02 2020 [Edit] [EN/CZ] Exponential formulas Formula Function domain ar.as=ar+s a^r . a^s = a^{r + s} ar.as=ar+s a∈R;r,s∈Na \in \R ; r,s \in \Na∈R;r,s∈N (ar)s=ars\left(a^r\right)^s = a^{rs}(ar)s=ars a∈R;r,s∈Na \in \R ; r,s \in \Na∈R;r,s∈N aras=ar−s\frac{a^r}{a^s} = a^{r - s}asar=ar−s a∈R−{0};r,s∈N;r>sa \in \R - \{0\}; r,s \in \N ; r > sa∈R−{0};r,s∈N;r>s (a.b)r=ar.br\left(a . b\right)^r = a^r . b^r(a.b)r=ar.br a,b∈R;r,s∈Na,b \in \R ; r,s \in \Na,b∈R;r,s∈N (ab)r=arbr\left(\frac{a}{b}\right)^r = \frac{a^r}{b^r}(ba)r=brar a∈R;b∈R−{0};r,s∈Na \in \R; b \in \R - \{0\} ; r,s \in \Na∈R;b∈R−{0};r,s∈N a0=1a^0 = 1a0=1 a∈R−{0}a \in \R - \{0\}a∈R−{0} 0m=00^m = 00m=0 m∈Nm \in \Nm∈N a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1 a∈R−{0};n∈Na \in \R - \{0\}; n \in \Na∈R−{0};n∈N (ab)−m=(ba)m\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m(ba)−m=(ab)m a,b∈R−{0};m∈Za,b \in \R - \{0\}; m \in \Za,b∈R−{0};m∈Z 1m1^m1m m∈Rm \in \Rm∈R a2n2n=∣a∣\sqrt[2n]{a^{2n}} = \lvert a \rvert2na2n=∣a∣ a∈R;n∈Na \in \R; n \in \Na∈R;n∈N (a)r=ar\left(\sqrt{a}\right)^r = \sqrt{a^r}(a)r=ar a≥0;r,s∈Ra \ge 0 ; r,s \in \Ra≥0;r,s∈R Logarithmic formulas Formula Function domain logaa=1\log_{a}a = 1logaa=1 a∈R+−{1}a \in \R^{+} - \{1\}a∈R+−{1} loga1=0\log_{a}1 = 0loga1=0 a∈R+−{1}a \in \R^{+} - \{1\}a∈R+−{1} logaar=r\log_{a}a^r = rlogaar=r a∈R+−{1};r∈Ra \in \R^{+} - \{1\} ;r \in \Ra∈R+−{1};r∈R logaxy=logax+logay\log_{a}xy = \log_{a}x + \log_{a}ylogaxy=logax+logay a∈R+−{1};x,y∈R+a \in \R^{+} - \{1\} ;x,y \in \R^{+}a∈R+−{1};x,y∈R+ logaxy=logax−logay\log_{a}\frac{x}{y} = \log_{a}x - \log_{a}ylogayx=logax−logay a∈R+−{1};x,y∈R+a \in \R^{+} - \{1\} ;x,y \in \R^{+}a∈R+−{1};x,y∈R+ logax=logbxlogba\log_{a}x = \frac{\log_{b}x}{\log_{b}a}logax=logbalogbx a,b∈R+−{1};x∈R+a,b \in \R^{+} - \{1\} ;x \in \R^{+}a,b∈R+−{1};x∈R+ logab=1logba\log_{a}b = \frac{1}{\log_{b}a}logab=logba1 a,b∈R+−{1}a,b \in \R^{+} - \{1\}a,b∈R+−{1}
