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.

you can use the derivative to show that the function is monotonic
the derivative is 
$ \frac{-1}{x^2} e ^{\frac{1}{x}} - \frac{1}{x} < 0$ 
so the function is monotonic
when x tends to 0 the function tends to ${\infty}$
and when x tends to ${\infty}$ the function tends to $-{\infty}$
so the function will get every real value exactly once
