For $C>3$ a constant, I'm interested in the roots of the function 
$$
f(x) = x - C \log x,
$$
defined for $x > 0$.
It's straightforward to say that this function has two roots by observing $f$ has a single local minimum at $x=C$ with $f(C) < 0$.
As $C \rightarrow \infty$, one of these roots approaches $1$, however, the second (larger) root grows at least linearly with $C$. Could the growth of this second root be bounded in terms of $C$?
In particular, I want to find an asymptotic upper bound for the larger root of $f$ as $C$ grows.

One easy bound is $e^C$, as $x > e^C$ implies $x > \log^2 x > C \log x$. Testing numerically, I feel this bound is very weak, but I have no idea how this could be improved or what the best possible bound could be.