What is the Solution for t in the Equation t^{t^2}=k?

[Gregg]

Homework Statement

Solve for t: $$t^{t^2}=k$$

Homework Equations

$$Y=Xe^X \iff X=W(Y)$$

The Attempt at a Solution

$$t^{t^2} = k$$

$$t = k^{1\over t^2}$$

$$t = e^{{1\over t^2} ln (k)}$$

$${1\over t} ln(k) = {1\over t^2} ln(k) e^{{1\over t^2} ln(k)}$$

$$t = \sqrt{ln(k)\over {W({{1\over t} ln(k)})}}$$

 

[Gregg]

This isn't right by the way. I'm wondering how to get to the right answers.

 

[Gregg]

$$t^{t^2} = k$$

$$t^{2{t^2}} = k^2$$

$$t^2= k^{2\over t^2}$$

$$t^2 = e^{{2\over t^2}ln(k)}$$

$$2ln(k) = {2ln(k)\over t^2}e^{{2\over t^2}ln(k)}$$

$$W(2ln(k)) = {2ln(k)\over t^2} \Rightarrow t = \pm \sqrt{2ln(k) \over W(2ln(k))}$$