Prove that `((log)_a N)/((log)_(a b)N)=1+(log)_a b`

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

If ((log)_a N)/((log)_c N)=((log)_a N-(log)_b N)/((log)_b N-(log)_c N), where N >0 and N!=1, a , b , c >0,!=1 , then prove that b^2=a c

Solve the system of equations: (log)_a x(log)_a(x y z)=48, (log)_a y log_a(x y z)=12 , a >0,\ a!=1(log)_a z log_a(x y z)=84\

If 3^x=4^(x-1) , then x= , (a) (2(log)_3 2)/(2(log)_3 2-1) (b) 2/(2-(log)_2 3) (c) 1/(1-(log)_4 3) (d) (2(log)_2 3)/(2(log)_2 3-1)

Compute the following a^(((log)_b((log)_a N))/((log)_b(log a)))

Given a^2+b^2=c^2, a >0 ; b >0; c >0,\ c-b!=1,\ c+b!=1,\ prove that : (log)_("c"+"b")a+(log)_("c"-"b")a=2(log)_("c"+"b")adot(log)_("c"-"b")a

If y^(x)= e^(y-x) , then prove that (dy)/(dx)= ((1+ log y)^(2))/(log y)

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 .

Statement I: If a=y^2,\ b=z^2, c=x^2,\ t h e n8(log)_a x^3dot(log)_b y^3dot(log)_c z^3=27 Statement II: (log)_b adot(log)_c b=(log)_c a ,\ also (log)_b a=1/("log"_a b) Statement 1 is True: Statement 2 is True; Statement 2 is a correct explanation for statement 1. Statement 1 is true, Statement 2 is true;2 Statement 2 not a correct explanation for statement 1. Statement 1 is true, statement 2 is false Statement 1 is false, statement 2 is true