If a, b, c are distinct positive real numbers in G.P and `log_ca, log_bc, log_ab` are in A.P, then find the common difference of this A.P

Explanation

Given that a,b,c are in G.P.,
Then `a=b/r,b=b,c=br`, where r is common ratio
Also given that `log_(c)a,log_(b)c,log_(a)b` are in A.P.
`rArrd=log_(b)c-log_(c)a=log_(a)b-log_(b)c`
`rArrd=(log_(10)c)/(log_(10)b)-(log_(10)a)/(log_(10)c)=(log_(10)b)/(log_(10)a)-(log_(10)c)/(log_(10)b)`
`rArrd=((log_(10)c)^(2)-log_(10)alog_(10)b)/(log_(10)blog_(10)c)`
`=((log_(10)b)^(2)-log_(10)alog_(10)c)/(log_(10)alog_(10)b)`
`rArrd=((log_(10)c)^(2)-log_(10)alog_(10)b-(log_(10)b)^(2)+log_(10)alog_(10)c)/(log_(10)blog_(10)c-log_(10)alog_(10)b)`
`rArrd=((logc-logb)(logc+logb+loga))/(logb(logc-loga))`
`=(logc/bcdotlogabc)/(logc/acdotlogb)`
`=(logr)/(logr^(2))cdot(logb^(3))/(logb)`
`=3/2`