First, you want to turn the x, y and z powers into factors. To do so, use the naperian logarithm:
color(gray)(Note: ln(a*b) = ln(a)+ln(b) and ln(a^b) = bln(a))
ln(2^x) = ln(3^y) = ln((24sqrt3)^z)
= xln(2) = yln(3) = zln(24sqrt3)
Then, isolate z:
z = (yln(3))/(ln(24sqrt3)) = (yln(3))/(ln(3*2^3)+ln(3^(1/2)))
= (yln(3))/(3ln(2)+ln(3)+ln(3)/2)
Replace ln(2) with y/xln(3)
z= (yln(3))/(3y/xln(3)+ln(3)+ln(3)/2)
= (ycancel(ln(3)))/((3y/x+1+1/2)cancel(ln(3))) = y/(3y/x+3/2)