Finding the intersections between $y = e^x$ and $y = x + 2$ algebraically?

Question

In trying to find the intersections between $y = e^x$ and $y = x + 2$ in terms of $x$, I came up with the equation,

$e^x = x + 2$

and subsequently,

$x = \ln(x+2)$.

Beyond that point, I am stumped. I am able to solve the equation numerically using a calculator, Newton's method, etc., but need to solve it algebraically. I have done a good deal of research on how to solve this type of problem, but have been unable to find any problems similar enough to be of help.

Thanks to the StackExchange community for your help. I love your sites and have been happy to find answers to hundreds of my own questions on them.

Answer 1

Here again appears the beautiful Lambert function : rewrite $$e^x=x+2\implies e^{x+2}=e^2(x+2)\implies e^y=e^2 y$$ and the solutions are given by $$x_1=-W\left(-\frac{1}{e^2}\right)-2$$ $$x_2=-W_{-1}\left(-\frac{1}{e^2}\right)-2$$ In fact, keep in mind that any equation which can write $A+Bx+C\log(D+Ex)=0$ has solutions in terms of Lambert function.

The Wikipedia page gives series approximations.

There is no other closed form to this equation. If you cannot use it, just numerical methods will provide the solutions.