<s>[INST] Is there a closed form for this transcendental equation? Wondering if there’s a closed form for $\exp(\frac{1}{\log(x)})=\frac{x}{e}$?
Numerically solving yields $x \approx .539$
 [/INST] $\exp \left (\dfrac{1}{\ln x} \right ) = \dfrac{x}{e}; \tag 1$
$\dfrac{1}{\ln x} = \ln \left ( \dfrac{x}{e} \right ) = \ln x - \ln e = \ln x - 1; \tag 2$
$1 = (\ln x)^2 - \ln x; \tag 3$
$(\ln x)^2 - \ln x - 1 = 0; \tag 4$
quadratic formula:
$\ln x = \dfrac{1 \pm \sqrt 5}{2}; \tag 5$
$x = \exp \left ( \dfrac{1 \pm \sqrt 5}{2} \right ).  \tag 6$
Some Additional Observations:
From (5), 
$\dfrac{1}{\ln x} = \dfrac{2}{1 \pm \sqrt 5} = \dfrac{2(1 \mp \sqrt 5)}{(1 \mp \sqrt 5)(1 \pm \sqrt 5}$
$= \dfrac{2(1 \mp \sqrt 5)}{1^2 - (\sqrt 5)^2} = \dfrac{2(1 \mp \sqrt 5)}{1 - 5} = \dfrac{2(1 \mp \sqrt 5)}{-4} = - \dfrac{1 \mp \sqrt 5}{2}; \tag 7$
and also from (5)
$\ln x - 1 = \dfrac{1 \pm \sqrt 5}{2} - 1 = \dfrac{-1 \pm \sqrt 5}{2} = - \dfrac{1 \mp \sqrt 5}{2} = \dfrac{1}{\ln x}; \tag 8$
from (6),
$\dfrac{x}{e} = xe^{-1} = \exp \left ( \dfrac{1 \pm \sqrt 5}{2} \right )e^{-1} = \exp \left ( \dfrac{1 \pm \sqrt 5}{2}  - 1 \right )$
$= \exp \left ( -\dfrac{1 \mp \sqrt 5}{2} \right ) = \exp \left ( \dfrac{1}{\ln x} \right ); \tag 9$
we also note that (4) yields
$\ln x( \ln x - 1) = 1 \Longrightarrow \dfrac{1}{\ln x} = \ln x - 1, \tag{10}$
in agreement with (8).
The reader is advised to take some care in keeping careful track of the $\pm$ and $\mp$ signs occurring in the above, since I have taken some liberty in their use.  But the intended meaning shouldn't be too hard to discern.
The reader may recall that the quadratic equation
$\phi^2 - \phi - 1 = 0, \tag{11}$
whose roots are of course
$\phi = \dfrac{1 \pm \sqrt 5}{2}, \tag{12}$
and obey
$\dfrac{1}{\phi} = \phi - 1, \tag{13}$
in fact quantifies the cassical golden section, which is the ratio of the sides of a rectangle such that if a square whose side is the shorter is removed the remaining rectangle is in the same proportion as the original.  So apparently what we're looking at here is the exponential/logarithmic version of that.  The further pursuit of this correspondence fascinates, but will be deferred to a later time.
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