How can we algebraically find the intersection of an exponential function y=a^x with the straight line y=x? In other words, what is the solution to the equation a^x = x ?
If , then put and the equation becomes:
which we can rearrange to:
and this has solution:
where is Lambert's W function.
(If we are not concerned with real solutions then the above works period, though you have to keep an eye on the multiple values of the W function)
RonLCode:This is EULER, Version 2.3 RL-06. Type help(Return) for help. Enter command: (16777216 Bytes free.) Processing configuration file. Done. >load "C:\Program Files\EulerRL\Euler Utils\LambertW.e"; > >a=0.1; >x=-LambertW(-log(a))/log(a) 0.399013 > > > >a^x-x 5.55112e-017 >
Some replies do not deserve an unthanked fate. For my money, the above two are in that class. Hence the pre-emptive safe-guard.
(Would've added rep but apparently I've got to spread a bit more first).
And this thread is certainly as good as any to direct the interested reader to the following references:
New analytic solution to classic problem?