If x,y,z,are in GP and  ax=by=cz, then

Given, x, y, z are in GP  and ax=by=cz Since,  x, y, z are  in GP.⇒ y2=xz ......(i)we have  ax=by=cz=k (say); k>0On taking  log both  sides,  we getxlog⁡a=ylog⁡b=zlog⁡c=log⁡k⇒ x=log⁡klog⁡a,y=log⁡klog⁡b,z=log⁡klog⁡cOn substituting  the  values of x,y and  z in Eq.  (i), we  get log⁡klog⁡b2=log⁡klog⁡a⋅log⁡klog⁡c⇒    (log⁡k)2(log⁡b)2=(log⁡k)2log⁡a⋅log⁡c⇒    (log⁡b)2=log⁡a⋅log⁡c⇒    log⁡alog⁡b=log⁡blog⁡c∴ logb⁡a=logc⁡b ∵loge⁡xloge⁡y=logy⁡x