First-Order Equations · Superchallenge problem

A continuously differentiable function $f$ on $[0,\infty)$ has $f(0)=2$ and $f'(x)>0$ for all $x>0$. For every $x>0$, the area under the graph of $f$ from $0$ to $x$ equals twice the arc length of the graph from $0$ to $x$. Find $f(2\ln3)$.