Suppose a,b,c>0,x,y,z>1 and a,b,c are in G.P. If alog⁡x=blog⁡y=clog⁡z then

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

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

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 $x, y, z .$

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