Given that log2(subscript) 3=x, log 2(subscript) 5=y and log2 (subscript) 7=z

Given that log2(subscript) 3=x, log 2(subscript) 5=y and log2 (subscript) 7=z, what is log2 (subscript) ((35^1/4)/(3^1/4)) expressed in terms of x, y, and z?

Well first off, you can take the ^(1/4) outside:

log(a^b) = b * log(a)
-->

¼log₂(35/3) = ¼ * { log₂(35) - log₂(3) } (because log(a/b) = log(a) - log(b))
-->

35 = 5*7, so log₂(35) = log₂(7*5) = log₂(7) + log₂(5) (because log(a*b) = log(a) + log(b))
-->

So putting that all back together we have:

¼log₂(35/3) = ¼ * { log₂(7) + log₂(5) - log₂(3) }
-->

so now plug in x, y and z

¼ * (z + y - x)
or
¼ * (y + z - x)
or
-¼ * (x - y - z)
...

log[2](3) = x
log[2](5) = y
log[2](7) = z


log[2](35^(1/4) / 3^(1/4) =>
log[2](35^(1/4)) - log[2](3^(1/4)) =>
(1/4) * log[2](35) - (1/4) * log[2](3) =>
(1/4) * (log[2](5) + log[2](7)) - (1/4) * log[2](3) =>
(1/4) * (log[2](5) + log[2](7) - log[2](3)) =>
(1/4) * (y + z - x)