Who can solve ln x = 1/x? I can get an approximate solution to any degree of accuracy, but is there an exact solution?
Let's just use Newton's method. It's always a good standby.
Let's make an initial guess of 1.5:
Iterate:
Keep going until you reach the desired accuracy.
The solution is 1.76291539065....
As far as an exact solution, I do not think so.
is pretty close of an approximation.
The exact solution can be written in terms of the Lambert W-function:
.
Therefore where W is the Lambert W-function.
W(1) is the Omega constant: http://en.wikipedia.org/wiki/Omega_constant.