Laws of indices

$$a^m \times a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}$$ $$a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad a^{\frac{m}{n}} = \sqrt[n]{a^m} \qquad (ab)^n = a^n b^n$$

Laws of logarithms

If $a^x = N$ then $\log_a N = x$. $$\log_a (xy) = \log_a x + \log_a y \qquad \log_a \frac{x}{y} = \log_a x - \log_a y$$ $$\log_a x^n = n\log_a x \qquad \log_a a = 1 \qquad \log_a 1 = 0$$ $$\log_a x = \frac{\log_b x}{\log_b a} \quad\text{(change of base)}$$