If `0ltxlt1`, prove that: `log(1+x)+log(1+x^2)+log(1+x^4)+… oo=-log(1-x)`

Prove that ln(1+x) 0.

(log x)^(log x),x gt1

(1+log x)^(2)/(x)

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

If higher powers of x^(2) are neglected, then the value of log(1+x^(2))-log(1+x)-log(1-x)=

2log x-log(x+1)-log(x-1) is equals to

Solve for x : 3^(log x)-2^(log x) =2^(log x+1)-3^(log x-1)

If (1 + 3 + 5 + .... " upto n terms ")/(4 + 7 + 10 + ... " upto n terms") = (20)/(7 " log"_(10)x) and n = log_(10)x + log_(10) x^((1)/(2)) + log_(10) x^((1)/(4)) + log_(10) x^((1)/(8)) + ... + oo , then x is equal to

f(x) = log(x^(2)+2)-log3 in [-1,1]

Prove that : int_(0)^(oo) log (x+(1)/(x)). (dx)/(1+x^(2)) = pi log_(e) 2