The properties that were used to derive the properties of logarithms are the properties of exponent because logarithms are exponents. The properties of exponents are: product of powers, power to a power, quotient of powers, power f a product and power of a quotient.
As an example, the log property log(a^k) = k log (a) can be derived from the exponential property (b^a)^k = b^(ak).
Likewise,
log (ab) = log (a) + log (b) comes from c^(a+b) = c^a*c^b
Proof:
Let x = c^a and y=c^b
Then,
log (x) = a and log (y) = b (base c)
log (xy) = log (c^a * c^b) = log (c^(a+b)) = a+b = log(x) + log (y)