Prove that `t a n(i(log)_e((a-i b)/(a+i b)))=(2a b)/(a^2-b^2)(w h e r ea ,b in R^+)`

Prove that tan(i log_(e)((a-ib)/(a+ib)))=(2ab)/(a^(2)-b^(2)) (where a,b in R^(+))

The value of a^(sqrt((log)_a b))-b^(sqrt((log)_b a))(w h e r ea ,b >0a n da , b!=1) is equal to...........

Solve the system the equations (a x)^(loga)=(b y)^(logb); b^(logx)=a^(logy) w h e r e a >0, b >0a n d a!=b , a b!=1

If log_e(1/(2)(a+b))=1/2(log_e a+log_e b), then a/(3b) =

Prove that a^x-b^y=0 w h e r e x=" "sqrt(("log")_a b) & y=sqrt((log)_b a ), a >0, b >0 & a , b!=1

prove that | vec axx vec b|=( vec adot vec b)t a ntheta, w h e r e theta is the angle between vec a a n d vec bdot

If a, b, c, d, e be 5 numbers such that a, b, c are in AP, b, c, d, are in GP & c, d, e are in HP then (i) Prove that a,c,e are in GP. (ii) Prove that e=(2b-a)^2//a . (iii) If a=2& e=18 , find all possible vlaues of b,c,d.

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

Comprehension 1 Let a^((log)_b x)=c w h e r e a , b , c & x are parameters. On the basis of above information, answer the following questions: If a=b=2xx3^sin^2theta+1/(1+tan^(2theta)) & c=(4 t a ntheta)/(sqrt(sec^2theta-1))(w h e r e theta lies in Ist Quadrant) , then x is- a.3 b. -3 c. 0 d. 4

Suppose A, B, C are defined as A=a^2b+a b^2-a^2c-a c^2, B=b^2c+b c^2-a^2b-a b^2, a n dC=a^2c+a c^2-b^2c-b c^2, w h e r ea > b > c >0 and the equation A x^2+B x+C=0 has equal roots, then a ,b ,c are in AdotPdot b. GdotPdot c. HdotPdot d. AdotGdotPdot