If `a^x=b ,\ b^y=c\ a n d\ c^z=a ,` prove that `x y z=1`

If a^(x)=b,b^(y)=c and c^(z)=a, prove that xyz=1

If a^x=b^y=c^z\ a n d\ b^2=a c , prove that y=(2x z)/(x+z)

If a^(x)=b^(y)=c^(z) and b^(2)=ac prove that (1)/(x)+(1)/(z)=(2)/(y)

If x=1+(log)_a b c ,\ y=1+(log)_b c a\ a n d\ z=1+(log)_c a b ,\ then prove that x y z=x y+y z+z xdot

If x=1+(log)_a b c , y=1+(log)_b c a and z=1+(log)_c a b , then prove that x y z=x y+y z+z x

If x=b-c+a, y=c-a+b, z=a-b+c , then prove that (b-a) x + (c-b)y +(a-c)z=0

If x ,\ y , z are not all zero and if a x+b y+c z=0; b x+c y+a z=0; c x+a y+b z=0 Prove that x : y : z=1:1:1\ or\ 1:omega:omega^2or\ 1:omega^2:omegadot

If a, b, c are in geometric progression, and if a^(1/x)=b^(1/y)=c^(1/z) , then prove that x, y, z are in arithmetic progression.

a^(x)=b^(y)=c^(z) and b^(2)=ac then prove that (1)/(x)+(1)/(z)=(2)/(y)

If x+z = 2y " and " b^2 = ac , then prove that a^(y-z)*b^(z-x)*c^(x-y) =1 .