Why is the following function surjective(onto) ?
$\displaystyle f: (1,\infty)\longrightarrow (\ln 2,\infty)$
$\displaystyle f(x)=\int_{x^2}^{x^4}\! \frac{1}{\ln t}\, dt$
The interemdiate value theorem says that if f(x) is continous on the interval [a, b] then it takes on all values between f(a) and f(b) on that interval.
Here, f(x) is $\displaystyle \int_{x^2}^{x^4}\frac{1}{ln t} dt$ which is continous on [tex][1, \infty) because ln t is continuous for all positive t and the integral of a continuous function is continuous.
You will still need to show that f(x) is unbounded. That is, that if y is any positive number there exist x such that f(x)> y. That, I think, is why Drexel28 mentions "some nice inequalities".