## How to solve a transcendental equation?

What is the best way to solve an equation of the form $x ^ x = 5$ by hand?

I know about approximations using Newton's method etc using a computer, but what I want to know is the best method to converge at a solution just using a pen and a paper.

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Wouldn't this entail computing an approximate value for the Lambert W-function? – Rob Grey Jun 6 2010 at 3:27
I retagged, as the "associative algebra" tag was confusing. Take the logarithm and use a pen and paper (as you like) to find the root of $\ln x=5/x$ graphically. Otherwise, follow the above comment. – Wadim Zudilin Jun 6 2010 at 3:33
With pen and paper I'd use log tables and Newton's method. – Dan Piponi Jun 6 2010 at 4:12
Sometimes the dumbest method has virtues... Just trying x^x on my calculator and using linear interpolation in my head I converged on 2.129372... really fast. It makes sense that if one used log tables and did the same thing with x log x = log 5 the same thing would happen, assuming you had enough digits of accuracy. – Michael Greenblatt Jun 6 2010 at 16:30
db, the examples in the wiki article, en.wikipedia.org/wiki/Lambert_function may be of interest for you – Pietro Majer Jun 6 2010 at 20:29