Find the smallest real number $a\gt 0$ for which the equation $a^x=x$ has no real solutions As the title says, 

We seek the smallest real number $a\gt 0$ for which the equation $a^x=x$ has no real solutions.

This is inspired by this question.
I must admit that I did not have much luck with this..Any suggestions?
 A: This is my solution, which I can't guarantee is correct or as rigorous as one would like.
First note that for $a \le 1$ it is trivial to see that real solutions exist, so we can focus on the $a > 1$ case. 
The idea then is to determine when the curves $a^x$ and $x$ intersects, which is exactly when the equation has solutions. So first reformulate it as a function and calculate it's derivatives:
$$
f(x) = a^x - x \\
f'(x) = ln(a)a^x - 1\\
f''(x) = ln(a)^2a^x
$$
Then use the derivatives to calculate $x_{min}$ for $f$, which is just number crunching and you get:
$$
-\frac{\ln(\ln a)}{\ln a}
$$
Then insert it into $f$ to get its $y_{min}$:
$$
\frac{1 + \ln(\ln a)}{\ln a}
$$
So if $y_{min} > 0$ the curves does not interest and no real solutions exist.
$$
\frac{1 + \ln(\ln a)}{\ln a} = 0 \\
\iff \ln(\ln a) = -1
\iff a = e^\frac{1}{e}
$$
In other words, solutions exist iff $0 < a \leq e^\frac{1}{e}$.
A: Obviously, $x\ge0$. From the equation, we draw
$$a=\sqrt[x]x.$$
This function has a single maximum at $x=e$, so that
$$a\le \sqrt[e]e.$$


A: As pointed out in the comments, this solution is not quite accurate. See Björn Lindqvist's solution.


Let $f(x) = a^x$. The idea is that we need to find the value of $x$ such that $f'(x) = 1$ and this point $x$ equals $a^x.$ The derivative is $f'(x) = \ln(a)a^x$. Setting this equal to one and solving for $x$ we see that 
$$
x = -\ln(\ln(a))\;.
$$ 
So our point is $(-\ln(\ln(a)),a^{-\ln(\ln(a))})$, and we need to lie on the line $y=x$, so we need the solution to the equation 
$$
-\ln(\ln(a))\;\;=\;\;a^{-\ln(\ln(a))}\;.
$$ 
I'm not sure if there is a way to solve for $a$ explicitly, but putting this into WolframAlpha we get the approximation $a \approx 1.27627610348955$. WolframAlpha, OEIS, and a brief Google search indicate that this isn't some known (or popular) constant.
