Given any $4$ numbers, is there a way to make $24$?

Question

I was playing the game $24$, and I saw that some numbers, such as $5,8,9,10$, could not be multiplied, divided, subtracted, added, etc. by me to get $24$ no matter how hard I tried...

Question: So that got me thinking: Is there a way to tell if a set of $4$ numbers can be manipulated to make $24$?

You can use any operation sign and any operation you would like (meaning you can use $\log$ and derivatives, but using those is probably going to be inefficient).

Answer 1

If we are allowed to use the Gamma function:

$$\Gamma(5)\cdot(9-8)^{10} = 24$$

or if we can use the factorial and square root:

$$\left(8^{\frac{10}{5\sqrt{9}}}\right)!$$

I can probably think up some more using other special-ish functions. The way I learned $24$ we could use addition, subtraction, multiplication, division, exponentiation, and square roots (and, of course, parentheses). I'm pretty sure there isn't a way to solve it with only those rules, although I'll keep trying.

Edit: We have

$$\sqrt{\left(8\sqrt{9}\right)^{\frac{10}{5}}} = 24$$

which uses only the "expanded standard" rules. It is the "cleanest" in my opinion of the solutions here.