How to show that this function gets every real value exactly once? $$f(x) = {e^{\frac{1}{x}}} - \ln x$$
I thought maybe to use the Intermediate value theorem.
Thought to create 2 functions from this one. and subtracting.
And finally I will get something in the form of 
$$f(c) - c = 0$$
$$f(c) = c$$
But how I proof that this function gets every real value but only once ?
So it must be monotonic.
What theorem should I use here ? 
Thanks.

Hint: set $z=\frac{1}{x}$. 
Then $f(x) = e^{z}+\log z = g(z)$ is a continuous increasing function, since it is the sum of two continuous increasing functions. Moreover: $$\lim_{z\to 0^+}g(z)=-\infty,\qquad \lim_{z\to +\infty} g(z) = +\infty. $$
