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

Question

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?

Answer 1

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 $0 < a \leq e^\frac{1}{e}$.

Answer 2

Obviously, $x\ge0$. From the equation, we draw

$$a=\sqrt[x]x.$$

This function has a single maximum at $x=e$, so that

$$a\le \sqrt[e]e.$$

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Answer 3

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

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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.