Name the properties that were used to derive the properties of logarithms.

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)