The expression `q^((sqrt(6+sqrt(6+sqrt(6+....oo))))^(log_3 p^([(log_q(log_q r))/(log_q p)]`

The expression x=log_(2)log_(9)sqrt(6+sqrt(6+sqrt(6+.........oo))) simplifies to

4^(5log_(4)(sqrt(2)(3-sqrt(6)))-6log_(8)(sqrt(3)-sqrt(2)))

Prove that (log3sqrt(3)+log sqrt(8)-log sqrt(125))/(log6-log5)=(3)/(2)

Find the value of (log sqrt(27) + log sqrt(8) - log sqrt(125))/(log 6 - log 5)

Prove that : (v) p^(log_(x)q) = q^(log_(x)p)

In the quadratic equation A(sqrt(3)-sqrt(2))x^(2)+(B)/(sqrt(3)+sqrt(2))x+C=0 with alpha,beta as its roots.If A=(49+20sqrt(6))^((1)/(4));B= sum of the infinit G.P. as 8sqrt(3)+8(sqrt(6))/(sqrt(3))+(16)/(sqrt(3))+........oo and | alpha-beta|=(6sqrt(6))^(k)k=log_(6)sqrt(10)-2log_(6)sqrt(5)+log_(6)sqrt(log_(6)18+log_(6)72) then find the value of C.

If log_p(q)+log_q(r)+log_r(p) vanishes where p,q,r are positive reals different than unity then the value of (log_p(q))^3+(log_q(r))^3+(log_r(p))^3

log_(a)((P)/(Q))=log_(a)P-log_(a)Q

If P =3^sqrt(log_(3)2)-2^(sqrt(log_(2)3))and Q=log_(2)log_(3)log_(2)log _(sqrt(3))81, then

If P =3^sqrt(log_(3)2)-2^(sqrt(log_(2)3))and Q=log_(2)log_(3)log_(2)log _(sqrt(3))81, then