is there any way to solve for x is $(e^x)/x =y$?

Question

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?

Answer 1

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 $$ e^x/x =y $$ $$ x e^{-x} = 1/y$$ $$ -x e^{-x} = -1/y$$ $$ -x = W(-1/ y)$$ $$ x=-W(-1/y)$$

Answer 2

$$y=\frac{e^x}{x}$$ 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: $$x=-W\left(-\frac 1y\right)$$

Answer 3

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.