The Basic log equation

log_x a = K where x^K = a

So that:

log₁₀ 10,000 = K or 10^K = 10,000 and K = 4

Basic Operations on logs of the same base

Addition

log_x y + log_x z = log_x (y * z)

Subtraction

log_x y - log_x z = log_x (y / z)

Multiplication

log_x y * log_x z = log_x (y^(log_xz))

Division

log_x y / log_x z = log_x (y^(1/(log_xz)))

Algebraic Operations for Base and Mantissa conversion

Most usefully: log_x y / log_x z = log_z y

log₁₀ 16 / log₁₀ 2 = log₂ 16

Also:

log_x y => log_z y * K

Such that: log_x y = log_x y * log_y z * log_z y

Further: 1 = log_y z * log_z y