There are many different ways to prove that log
a
(xy) =
log
a
(x) +
log
a
(y)
. Let
z=
log
a
(xy)
,
k1=
log
a
(x), k2=
log
a
(y)
. Then
a
z
=xy
,
a
k1
=x
,
a
k2
=y
Since
a
k1+k2
=a
k1
a
k2
=xy
, therefore the equation holds.