e^x = x
since e^x is always positive, there are no negative roots.
Since the slope of y=x is 1, and the slope of y=e^x is greater than 1 for positive x, then since e^0>0, the two graphs never intersect.
In short, there are no real solutions.
Positive and negative roots of e to the power x equals to x.
2 answers
Confirming Steve's conclusion:
https://www.wolframalpha.com/input/?i=Plot+y+%3D+e%5Ex,+y+%3D+x
https://www.wolframalpha.com/input/?i=Plot+y+%3D+e%5Ex,+y+%3D+x