I'm trying to solve a bigger problem however I am stuck at this step:
How can I solve:
$$ 2^x - x = 5 $$
any hints/tips/steps please?
As alluded to in the comments there is an integer solution. For the other solution is existence of a solution good enough? You can use the Intermediate Value Theorem on the function $f(x)=2^x-x-5$. It is negative at $x=0$ and positive at $x=-6$. So, somewhere in between the IVT says there must be a $0$. Or you can use Newton's method on $f$ to approximate the $0$ of $f$.