is this conjecture true or false?

Question

I want to know if this conjecture istrue or false

$$\Large e^{\frac{ \ln x}{x}} \notin \mathbb{Z} $$

for every $x \in \mathbb{R} \setminus \{1,-1,0\} $

Answer 1

You should ask this only for $x>0$, as the expression is not well defined otherwise. You can rule out the case $x\in(0,1)$ easily, since it implies $\ln(x)/x<0$. Now find the maximum of $\ln(x)/x$ on $(1,\infty)$, and conclude.

Answer 2

Hint
Firstly, show that as a real function, ${\rm dom}(f) = \mathbb R^{>0}$
Secondly, show that $f(\mathbb R^{>0}) = (0, e^{\frac1e})$ and conclude by $$f(\mathbb R^{>0}) \cap \mathbb Z = \{1\}$$ and $f^{-1}(1) = \{1\}$

Answer 3

Just like Oliver Bel said, $e^{lnx/x} = x^{1/x}$.

Now, suppose there exist an integer N such that $N^x=x, N>1$. Try to graph the equations $y = N^x$ and $y = x$. It should be clear that no such integer exists.

Answer 4

$x^{1/x}$ takes the value 1 at 1, and decreases after $e$, since its derivative is:

$-x^{\frac{1}{x}-2}(logx-1) $. And $-x^{\frac{1}{x}-2}$ is always $<0$ , and $logx-1>0$ is positive for $x>e$. So we only need to worry about $e^{1/e}~1.44....<2$, all computations thanks to Wolfram. So only possible integer value in $[1,\infty)$ is $1$