Solve the Equation:

Question

The equation is: $$x^{xn} = n $$ I continued by taking logarithm on both sides but end up with an equation of the form: transcendental=rational, which I was unable to solve!!

Answer 1

To solve for $x$ I began by raising both sites to $\frac{1}{n}$-th power: $$x^x=n^\frac{1}{n}$$ Taking log on both sides gives: $$x\ln(x)=\frac{1}{n}\ln(n)$$ Now let $u=\ln(x), \ x=e^u \ $ $$e^u u=\frac{1}{n}\ln(n)$$ From the definition of Lambert W function $$u=\text{W}(\frac{1}{n}\ln(n))$$ Resubstituting gives: $$\ln(x)=\text{W}(\frac{1}{n}\ln(n))$$ $$x=\exp(\text{W}(\frac{1}{n}\ln(n)))$$ Which is solution to your equation. I hope that I've done everything properly.