Prove that `ln(1+x) < x` for `x > 0.`

Prove that ln (1+1/x) gt (1)/(1+x), x gt 0 . Hence, show that the function f(x)=(1+1/x)^(x) strictly increases in (0, oo) .

If log_(1/3) |z+1| > log_(1/3) |z-1| then prove that Re(z) < 0.

Prove that ln 2 lt int_(0)^(1)(dx)/(sqrt(1+x^(4))) lt (pi)/2)

Let f(x) ={{:((cos""(1)/(x))(log(1+x))^(2),xge0),(0,xle0):} . Prove that f(x) if differentiable but derivative is not continuous at x=0.

(i) If log(a + 1) = log(4a - 3) - log 3, find a. (ii) If 2 log y - log x - 3 = 0, express x in termss of y. (iii) Prove that : log_(10) 125 = 3(1 - log_(10)2) .

Assuming that log (mn) = log m + logn prove that log x^(n) = n log x, n in N

Prove that : log _ y^(x) . log _z^(y) . log _a^(z) = log _a ^(x)

Prove that: log sin 8x = 3 log2+log sin x+ log cos 2x+log cos 4x

Prove that log_(e)(1+x)ltxforxgt0

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