x ^a x ^b = x ^{(a + b)}

x ^a y ^a = (xy) ^a

(x ^a) ^b = x ^(ab)

x ^(a/b) = b^th root of (x ^a) = ( b^th (x) ) ^a

x ^(-a) = 1 / x ^a

x ^{(a - b)} = x ^a / x ^b

Logarithms

log_b(1) = 0

log_b(b) = 1

log_b(x*y) = log_b(x) + log_b(y)

log_b(x/y) = log_b(x) - log_b(y)

log_b(x ⁿ) = n log_b(x)