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

If a >1,b >1 and c >1 are in G.P. then show that 1/(1+(log)_e a),1/(1+(log)_e b), a n d1/(1+(log)_e c) are in H.P.

The differential coefficient of f((log)_e x) with respect to x , where f(x)=(log)_e x , is (a) x/((log)_e x) (b) 1/x(log)_e x (c) 1/(x(log)_e x) (d) none of these

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

(1)/(x log x log (log x ))

Prove log_(b) 1=0

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

If x ,y ,z are in G.P. and a^x=b^y=c^z , then (log)_b a=(log)_a c b. (log)_c b=(log)_a c c. (log)_b a=(log)_c b d. none of these

The value of (1+2(log)_3 2)/((1+(log)_3 2)^2)+((log)_6 2)^2 is 2 (b) 3 (c) 4 (d) 1

The value of (6a^((log)_e b)((log)_(a^2)b)((log)_(b^2)a))/(e^((log)_e a(log)_e b)) is (a) independent of a (b) independent of b (c) dependent on a (d) dependent on b

T h ev a l u eofint_1^((1+sqrt(5))/2)(x^2+1)/(x^4-x^2+1)log(1+x-1/x)dxi s (a) pi/8(log)_e2 (b) pi/2(log)_e2 (c) -pi/2(log)_e2 (d) none of these