Is there a way to solve this equation? (maybe with Lambert's W Function?)

Question

I'd like to know if there is a way to solve the equation $$x\ln x=\alpha+\beta x$$ for known constants $\alpha,\beta\in\mathbb{R}$. I know that Lambert's W Function $W$ can be used to solve $$x\ln x=\alpha$$ because then $x=e^{W(\alpha )}$, but in the upper problem I can't reformulate the equation in a way that lets me use Lambert's W function.

Does anyone know how to do this? Or any way to find a solution for $x$?

Thanks.

Answer 1

Note that $x\ln(x) - x\beta = x\ln(xe^{-\beta})$, so we substitute $y=xe^{-\beta}$ and get $$y \ln(y) = \alpha e^{-\beta}.$$ Thus $y = \exp(W(\alpha e^{-\beta}))$ and $x = \exp(\beta+W(\alpha e^{-\beta}))$.

Answer 2

We have \begin{align*} x\log x &= \alpha x +\beta\\ \iff \beta &= x(\log x - \alpha)\\ &= x\log\bigl(x\exp(-\alpha)\bigr)\\ \iff \exp(-\alpha)\beta &= \exp(-\alpha)x \log\bigl(x\exp(-\alpha)\bigr)\\ \iff \exp(-\alpha)x &= \exp\bigl(W(\exp(-\alpha)\beta)\bigr) \end{align*}