Question

(log(x^6)+2log(y^3))/log(xy)

Answer 1

remember a*log(b)= log(b^a)
and log a+ log b= log (ab)

and log a/logb= log (a)-log (b)

Answer 2

recall that when the expression inside the log is raised to a certain power, we can write the exponent outside of the log, for instance:
log (x^2) = 2 log x
log [(2y)^0.5] = 0.5 log 2y
thus for the problem, we can rewrite it as:
(log(x^6) + 2log(y^3)) / log(xy)
(6 log x + 6 log y ) / log (xy)
6 (log x + log y) / log (xy)
now recall that the sum of log with the same base is the same as the log of the product of the expressions inside the parenthesis, for instance:
log 2 + log x = log 2x
log a + log b = log ab
thus we can further simplify this as:
6 (log x + log y) / log (xy)
6 (log xy) / log xy
cancelling out the log terms, we have 6

hope this helps~ :)