I'm trying to find where these two functions meet (lg meaning log base 2), and by continuously fiddling with the calculator i found that when x = 981,970001342... they are equal, but is this the only one? And where does it come from properly, not by guesses.
I tried doing (lg(x))^3 = x (g(x) = f(x)), but i have not been able to find a solution, probably due to my not so good mathematical background.
You won't solve this analytically without the Lambert W function. There are many algorithms to find the roots of a function, but they all come down to choosing $x$ values (guided by what you know about the function, including the previous results) and seeing if the function value is small. There is another root at about $2.59092475852820$. A plot of $\lg^3 x -x$ from $2$ to $1000$ is below. You can see there are two roots. You can prove there are no more by showing the derivative does not change sign beyond what you can see.