Find $x$ such that $5^x – \sqrt{2x} -\log_2{x} = 22$

Find $ x$ such that $ 5^x – \sqrt{2x} -\log_2{x} = 22$ .

I have observed that the solution of this equation should be $ x = 2$ , I also plotted the graph of the function $ f(x) = 5^x – \sqrt{2x} -\log_2{x}$ , and it looks like an increasing function, so the solution should be unique.

However my problem is proving that this solution is unique, and showing that the function is increasing using its derivative does not seem to work, as the expression is pretty ugly.

Do you have any suggestions on how to solve this without using the derivative?