`1/(1+log_b a+log_b c)+1/(1+log_c a+log_c b)+1/(1+log_a b+log_a c)`

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

Q. If log_x a, a^(x/2) and log_b x are in G.P. then x is equal to (1) log_a(log_b a) (2) log_a(log_e a)+log_a log_b b (3) -log_a(log_a b) (4) none of these

If log_a a. log_c a +log_c b. log_a b + log_a c. log_b c=3 (where a, b, c are different positive real nu then find the value of a bc.

If (log)_b a(log)_c a+(log)_a b(log)_c b+(log)_a c(log)_bc=3 (where a , b , c are different positive real numbers !=1), then find the value of a b c dot

Prove the following identities: (a) (log_(a) n)/(log_(ab) n) = 1+ log_(a) b" "(b) log_(ab) x = (log_(a) x log_(b) x)/(log_(a) x + log_(b) x) .

Simplify: 1/(1+(log)_a b c)+1/(1+(log)_b c a)+1/(1+(log)_c a b)

Simplify: 1/(1+(log)_a b c)+1/(1+(log)_b c a)+1/(1+(log)_c a b)

If ((log)_a N)/((log)_c N)=((log)_a N-(log)_b N)/((log)_b N-(log)_c N),w h e r eN >0a n dN!=1, a , b , c >0 and not equal to 1, then prove that b^2=a c

Prove the identity; (log)_a Ndot(log)_b N+(log)_b Ndot(log)_c N+(log)_c Ndot(log)_a N=((log)_a Ndot(log)_b Ndot(log)_c N)/((log)_(a b c)N)

Prove that : (1)/( log _(a) ^(abc) )+(1)/(log_(b)^(abc) ) + (1)/( log _c^(abc)) =1