I used the Newton-Raphson method to find the root of the function f(X)=exp(aX)-X, where a=.5*log(2/86400)=-5.336798 approximately.
This yields the iteration Xn+1=Xn-f(Xn)/f'(Xn) which converges very rapidly to a value of X=0.25560605642100. Starting at X0=0, after 1 iteration, the estimate is in error by about 9.8e-2, after 2 iterations, the estimate is in error by about 1.5e-2, after 3 iterations the estimate is in error by about 3.5e-4, after 4 iterations the estimate is in error by 1.9e-7 and after 5 iterations it has has converged to an estimate that is in error by 5.6e-14.
Is there a better way to do this?
UGH!!! Thx for the effort (I was the one to post that question, as a part of a solution to another problem I found here on Yahoo! Answers.) Incidentally, I made a mistake when simplifying the other problem and it should be xln(2)-2ln(x)-ln(86400) = 0
I don't know any way to get rid of the ln(x) unless maybe it's using calculus techniques?