If `a, b, c` are distinct positive real numbers each different from unity such that `(log_aa.log_c a -log_a a) + (log_a b.log_c b-logb_ b) + (log_a c.log_a c - log_c c) = 0,` then prove that `abc = 1.`

If a,b,c are distinct positive real numbers each different from unity such that (log_(a)a.log_(c)a-log_(a)a)+(log_(a)b*log_(c)b-log b_(b))+(log_(a)c.log_(a)c-log_(c)c)=0 then prove that abc=1

If a,b,c are distinct real number different from 1 such that (log_(b)a. log_(c)a-log_(a)a) + (log_(a)b.log_(c)b.log_(c)b-log_(b)b) +(log_(a)c.log_(b)c-log_(c)C)=0 , then abc is equal to

If x is a positive real number different from unity such that 2log_(b)x=log_(a)x+log_(c)x then prove that c^(2)=(ca)^(log_(a)b)

Show that log_(b)a log_(c)b log_(a)c=1

Let a,b" and "c are distinct positive numbers,none of them is equal to unity such that log _(b)a .log_(c)a+log_(a)b*log_(c)b+log_(a)c*log_(b)c-log_(b)a sqrt(a)*log_(sqrt(c))b^(1/3)*log_(a)c^(3)=0, then the value of abc is -

(1+log_(c)a)log_(a)x*log_(b)c=log_(b)x log_(a)x