>>13487920>do my homeworkx ln (x) (ln (x ln (x))) = 0 gives 3 possibilities:
1) x = 0
2) ln(x) = 0 => x = e^0 = 1
3) (ln (x ln (x))) = 0 =>
=> (x ln (x)) = 1 =>
=> ln(x^x) = 1 =>
x^x = e => x = 1.763....
Writing (x ln (x)) = 1 we get ln(x) * e^(ln(x)) = 1,
therefore, denoting ln(x) = z, our equation is z * e^z = 1. We get
z = ln(x) = W(1), where W is the Lambert W function:
https://en.wikipedia.org/wiki/Lambert_W_functionSo the exact solution is e^W(1).
https://www.wolframalpha.com/input/?i=e%5EW%281%29Tough most systems don't implement Lambert functions, so you'd be better off using Newton's method or such to extract the numerical solution out of x^x = e.