How to solve one unknown equations like $15^x = x^{16}$ or $x^x = 1991$ etc...?

Question

I am curious how to solve equations like $15^x = x^{16}$ or $x^x = 1991$. Yes, of course Wolfram Alpha or Matlab can calculate it, or it can be approximated (as we learned in school). But are there any other solution methods? They can be transformed to a form where on one side there is $x\ln(x)$ or $x/\ln(x)$. How to solve them? What if we have $15^x+16^x+x^x-x^{17}-x^{18} = 0$? ($x>0$ in all equations).

Answer 1

One option is to find the solution in terms of Lambert $W$ functions. Lambert $W$ function is defined as: $$ W(x)e^{W(x)}=x. $$ Now for $15^x=x^{16}$, you can make an equivalent equation which is: $$ 15^{\frac{-x}{16}}x=1 $$ $$ e^{x\frac{-\ln15}{16}}\left(x\frac{-\ln15}{16}\right)=\frac{-\ln15}{16} $$ And then you get: $$ x=\frac{-16}{\ln15}W(\frac{-\ln15}{16}). $$ For $x^x=1991$, as you said it yourself, you can turn it into:

$$ e^{\ln x}\ln x=\ln(1991)\rightarrow x=e^{W\left(\ln1991\right)}. $$ Now for the case of sum of polynomials and mixture of exponential function, the current form of Lambert $W$ function will not work. Look at the generalization of Lambert function in Wikipedia.