Suppose $a,b,c>0,x,y,z>1$ and $a,b,c$ are in G.P. If $a^{\log x}=b^{\log y}=c^{\log z}$ then 

$a^{\log x}=b^{\log y}=c^{\log z}=k\text{ (say) }$

$\Rightarrow a=k^{1/\log x},b=k^{1/\log y},c=k^{1/\log z}$

Now, $a, b, c$are in G.P.

$\Rightarrow \frac{1}{\log x},\frac{1}{\log y},\frac{1}{\log z}$ are in A.P. 

$\Rightarrow \log x,\log y,\log z$ are in H.P.

However, nothing can be said about $x, y, z.$