simple problem

Dear /sci/,

I cannot figure out how to find n in the equation n^n = 10^9. Explanations would be excellent.

Love,
     Anonymous.

I've stumbled upon a page that takes equations, uses Mathematica, and returns results.  The solutions of such equations make use of something called the Lambert W function, which is also called ProductLog.  When given a value V, the function gives you the value w needed for V = w*e^w to be true. e is the base associated with natural logarithm.

http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=basic
http://en.wikipedia.org/wiki/Lambert's_W_function

From the wiki page, I see that your n = ln(10^9)/W(ln(10^9)
From the solver page, I get n about 9.29509

woops, missed a bracket: n = ln(10^9)/W(ln(10^9))

>>1

Use the Newton-Raphson method.

The solution to 10^9 = n^n is not at all nice. The Lambert W function is your best bet.

square root

USE NUMERICAL ANALYSIS + COMPUTERS

anyway if you just want a number answer and don't need to write an equation that isolates n (see >>3), you can use the goal seek feature of Excel.