is there any way to solve for x is $(e^x)/x =y$? I have tried using all the functions I know of and have been unable to get anywhere. I know it is impossible to solve this equation using elementary operations. I know it is impossible to solve for x with elementary functions because if $y=4$ there are two values of x. I tried using logarithms and got $x-ln(x)=ln(y).$ I am still stuck, what should I do?
 A: $$y=\frac{e^x}{x}$$
then we can say:
$$xe^{-x}=\frac 1y$$
by letting $u=-x$ we get:
$$ue^u=-\frac 1y$$
Now we apply the Lambert W function to both sides:
$$u=W\left(-\frac 1y\right)$$
now undo the substitution:
$$x=-W\left(-\frac 1y\right)$$
A: For a general $y$ there's no way to express $x$ using elementary functions. In general you need to use the special function called Lambert W function.
You have $$ e^x/x  =y $$
$$ x e^{-x} = 1/y$$
$$ -x e^{-x} = -1/y$$
$$ -x = W(-1/ y)$$
$$ x=-W(-1/y)$$
A: You can use Newton's approximation method, by iteration.  In this particular case where $$y=\frac{e^x}{x}$$ the iteration will be: $$x_{n+1}=x_{n}-(x_{n}-e^x/Y)/(1-e^x/Y)$$ with $Y$ being the $y$ value for which we are trying to solve $x$.  The 'seed' value is $x_{1}$.  The iteration converges rapidly to both solutions of $x$ when $Y$=4.  For instance, for a seed value of $x_{1}=3$, $x$ converges to 2.15329236 (8 decimal places) in only 5 iterations.  For a seed value of $x_{1}=0.5$, $x$ converges to 0.357402956. The seed value interval must be so that the derivative of $$x-(x-e^x/Y)/(1-e^x/Y)$$ be $\le m <1$.  I have not checked out the details but it appears that this is true for a wide range of positive or negative $x$ values.
