The "ceiling" of , and so the right side, changes only when x is a power of two. If , . In order for that to be equal to we must have , .

You might be able to solve that algebraically using the Lambert W function but I would try a numerical method:

If n= 1, . If n= 2 we have . If n= 3 we have . If n= 4 we have . If n= 5 we have . That tells us that there must be a solution to for n a numberbetweenn= 4 and n= 5. Okay, try half way between (just because that is simplest). If n= 4.5, [tex]2^{2.5}= 4\sqrt{2}[/quote] which is just slightly more than 4.5+ 1= 4.6. There must be a solution between n= 4 and 4.5 (and so x between and [tex]2^{4.5}= 22.67...