solve x^x=10?

Question

How can u do this without newton's method. i mean how to find the exact solution.

and how do u find W(ln10)

Answer 1

If you are limited to just the functions you learned in high school, you cannot. If you have access to the Lambert W function, however, you may solve this equation as follows:

x^x=10

Take logs of both sides:

x ln x = ln 10

Represent x as e^(ln x)

ln x e^(ln x) = ln 10

Invoke the Lambert W function:

ln x = W(ln 10)

Exponentiate:

x = e^(W(ln 10))

And we are done.

Edit: It's implemented in Mathematica as ProductLog[x]. There's an online evaluator here: http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog

Answer 2

Basically this is taking the square root of 10 which will give you 3.1622776601683793319988935444327

Answer 3

e=2.7272727272