Does $ 2=5x+x\log x$ have a closed form solution?

Question

I tried to find the solution for the equation:

$$ 2=5x+x\log x, $$ which is equivalent to $$ \log x=\frac{2}{x}-5. $$

It doesn't seem to admit an exact solution. I only found that the numerical solution is $0.47$. Please help or give me some hints. Thanks.

Answer 1

With $\dfrac2x=y$, the equation $$2=5x+x\log x$$

turns to (after some rewrite)

$$ye^y=2e^5.$$

This is a Lambert equation, which proves that there is no closed-form solution (except using the ad-hoc function $W$).

Answer 2

Hint: It is $$x={{\rm e}^{{\rm W} \left(2\,{{\rm e}^{5}}\right)-5}}$$

Answer 3

First of all this equation has two real solutions:

$$ x \approx 146.399 $$ $$ x \approx 0.326899$$

this can be seen on the graph below, or by considering asymptotic behavior of both functions.