if log a= x and log b = y,

Question

then what is 1/3 log (ab)^3？

Answer 1

well first of all:
1/3 log (ab)^3
= (1/3)(3) log (ab) = log (ab)
now log (ab) = log a + log b
(that is how your slide rule works)
(you do not have a slide rule?)
(google slide rule :)

so what we have here is
x+y

Answer 2

http://en.wikipedia.org/wiki/Slide_rule

Answer 3

Now about that power stuff,

If we can agree that
log(ab) = log (a) + log (b)
then what is log (a^3) ????
well
log (a*a*a) = log(a)+log(a)+log(a)
which is of course
3 log (a)
:)