Base 10 log both sides: xlog(x)=100

graph y=xlogx and y=100 and see where they cross

https://www.desmos.com/calculator/tte3op9ayp

From which x=56.961

Harder way:

Use Lambert W function, the inverse of f(x)=x*e^x

Let x=e^t

then take the natural logarithm of both sides

100ln10 = t*e^t

t = W(100ln10) 

x= e^W(100ln10)

Implementing formula for this function is very messy (open the set-up tab in the link below.) I didn't make the initial Desmos graph, but I added to it to solve this problem

https://www.desmos.com/calculator/4voqqbelmc

From which x=56.9612484323

(WolframAlpha agrees with the above and can give even more precision.)