Let $W(z)$ be the solution of $z=We^W$, then the solution to the original equation is:
One trivial numerical solutionis $t(1.44)=1.44$.
$W(z)$ is the Lambert W Function. It is a build-in function in Mathematica(7.0) called ProductLog.
Here is a plot of $t(a)$ vs. $a$ (purple). It is mentioned that there is another solution near t=8.040854. We can see it from the figure. The horizontal blue line is $t=1.44$
There seemed to be some confusion about how many solutions the original equation (shown below) has:
Taking the $\log$ on both sides we can rewrite it as:
The first solution to (10) is obvious: $t=1.44$
Numerical result (by @Arthur) also showed that there exists a second solution: $t=8.40854$.
This is because:
The question then is: are there any other solutions to (10)?
I do not know how Arthur got the second numerical solution. By expressing the solution to (10) as in (2), we can just plot t(a) vs. a to see if there are any more unexpected solutions.
For example, we can say that the following equation (13) has no real solution for t:
All of this can only be achieved after we express the solution to (10) as in (2) and we carry out the numerical experiments based on the knowledge of Lambert W Function.
Now we can ask the following question.
How many solution of $t$ exist for equation (14)?
It is now obvious to us that there are two solutions: $t=1.44$ and $t=8.40854$.
This is why I made some comments before saying that the second solution to (10) has nothing to do with Lambert W Function. It also showed up in (14) with $\exp$ function.
Hope my explanation helps-