logarithms. find z?

Question

log_2x(z) = 3
log_5y(z) = 6
log_xy(z) = 2/3
Find z.

Answer 1

If I am reading your notation correctly (the first one is log base 2x of z is 3?) then you convert these into exponential equations to get:

z=(2x)^3
z=(5y)^6
z=(xy)^(2/3)

Then get a common multiple (like 6) for the x&y powers on the right:
square z^2=(2x)^6
nothing z=(5y)^6
ninth z^9=(xy)^6
Then take a ratio and all the x's and y's cancel:
z^2*z/z^9 = (2x)^6*(5y)^6/(xy)^6 = 2^6 * 5^6
z^(-6) = 10^6
take -1/6 powers of both sides:
z^1 = 10^-1 or z=1/10