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)
