Hi I recently posted a this question earlier and got some excellent answers but to take it a little further I liked k170's answer however it contained a Lambert W Function in the answer and I was wondering if I could get a more simple answer. Wolfram Alpha gave an approximate answer x=0.143668 I don't know how it got this answer but if anyone does it would be greatly appreciated.
Other Post Link:How to solve 5x=0.01^x
The Lambert W function solution is the best you can do analytically. There is no solution to your equation in elementary functions.
As for a numerical solution, you have all sorts of options. An easy-to-understand method is bisection. It should not be hard to see that there is exactly one solution, and that this solution is between $0$ and $1$ (just draw the graph if that's not clear).
So you can run bisection on $[0,1]$. Let's start. Note that $5(0)-0.01^0<0$ and $5(1)-0.01^1>0$. Now you consider $0.5$: you find $5(0.5)-0.01^{0.5}>0$. So you keep the left endpoint (the negative one) the same and replace the right endpoint (the positive one) with $0.5$. Now you repeat this procedure on $[0,0.5]$. If you want a number at the end, take the midpoint of the last interval that you find.