$${\log b^a = a \log b}$$
$${\log a + \log b = \log ab}$$
$${\log a - \log b = \log \frac{a}{b}}$$
$${\log_b 1 = 0}$$
$${\log_b a = \frac{log_c a}{log_c b}}$$
$${y = \log_b x \Rightarrow x = b^y}$$
$${b^{\log_b a} = a}$$
$${(a^b)^c = a^{bc} = (a^c)^b}$$
$${\frac{a^b}{c^b} = (\frac{a}{c})^b}$$
$${a^b c^b = (ac)^b}$$
$${a^b a^c = a^{c+b}}$$
$${x^2 - y^2 = (x+y)(x-y)}$$
$${x^3 - y^3 = (x-y)(x^2 + xy + y^2)}$$
$${x^3 + y^3 = (x+y)(x^2 - xy + y^2)}$$
$${x^n - 1 = (x - 1)(x^{n-1} + x^{n-2} + x^{n-3} + \cdots + 1)}$$
$${x^n - 1 = (x + 1)(x^{n-1} - x^{n-2} + x^{n-3} - x^{n-4} + \cdots - 1), n = 2k}$$
$${x^n + 1 = (x + 1)(x^{n-1} - x^{n-2} + x^{n-3} - x^{n-4} + \cdots + 1), n = 2k + 1}$$
$${(x + y)^2 = x^2 + 2xy + y^2}$$
$${(x - y)^2 = x^2 - 2xy + y^2}$$
$${(x + y)^3 = x^3 + 3xy(x+y) + y^3}$$
$${(x - y)^3 = x^3 - 3xy(x-y) - y^3}$$
$${(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + xz)}$$
$${(x + y + z + w)^2 = x^2 + y^2 + z^2 + w^2 + 2(xy + xz + xw + yz + yw + zw)}$$
$${(a^2 + b^2) (c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2 }$$
$${(a^2 + b^2) (c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 }$$
Interpretation: the product of 2 numbers that can be expressed as a sum of squares can also we expressed as a sum of 2 squares.
Notice that it also implies it can be expressed 2 ways assuming \(a \not = b\) and \(c \not = d\) and \(a, b \not = 0\).