Find exact solution of $(x+1) e^x = \frac 12$.

Question

I simplified an equation down to this: $$(x+1) e^x = \frac 12.$$

Then I am perplexed on how I can get an exact solution out of this. I can see graphically (split it on either side) and see where they cross which is around $-0.3$ or so.

I thought to use the identity $$\ln(AB)=\ln(A)+\ln(B)$$ to help, but it won't do much as I will still have $x$ and $\ln(x)$ in the equation: $x+\ln(x+1)=\ln(1/2)$ Which is still not intuitive. (if I apply Lawn of differences to this it will take me right back to where I started of course!)

Any cool tricks I can apply to help find an exact solution?

Answer 1

There isn't a solution in terms of common functions. You might notice that the presence of a term $x e^x$ would mean that you'd want to use the "Lambert W function", which is defined $f(\cdot)$, the principal solution of

$$f(w) \exp(f(w))= w.$$

Then, we have

$$w_1 + e^x = 1/2, w_1 = x e^x$$

or, more usefully,

$$w_2/e=1/2, w_2 = (x+1)e^{x+1}$$

Then, the solution is

$$x = W(e/2)-1.$$