$${\log b^a = a \log b}$$
$${\log a + \log b = \log ab}$$
$${\log a - \log b = \log \frac{a}{b}}$$
$${\log_b 1 = 0}$$
$${\log_b a = \frac{log_c a}{log_c b}}$$
$${y = \log_b x \Rightarrow x = b^y}$$
$${b^{\log_b a} = a}$$
$${(a^b)^c = a^{bc} = (a^c)^b}$$
$${\frac{a^b}{c^b} = (\frac{a}{c})^b}$$
$${a^b c^b = (ac)^b}$$
$${a^b a^c = a^{c+b}}$$
$${(x + y)^2 = x^2 + 2xy + y^2}$$
$${(x - y)^2 = x^2 - 2xy + y^2}$$
$${x^2 - y^2 = (x+y)(x-y)}$$
$${x^3 - y^3 = (x-y)(x^2 + xy + y^2)}$$
$${x^3 + y^3 = (x+y)(x^2 - xy + y^2)}$$
$${x^n - 1 = (x - 1)(x^{n-1} + x^{n-2} + x^{n-3} + \cdots + 1)}$$
$${x^n - 1 = (x + 1)(x^{n-1} - x^{n-2} + x^{n-3} - x^{n-4} + \cdots - 1), n = 2k}$$
$${x^n + 1 = (x + 1)(x^{n-1} - x^{n-2} + x^{n-3} - x^{n-4} + \cdots + 1), n = 2k + 1}$$