How to solve $x! = 10$

Question

Pretty much what the title says. To be more general, try to solve $x! = n$. I have tried for many hours and only ended up with a headache, is there any good/decent/practical way of solving such an eqation? I could not find anything about this on the internet.

Tl;dr What would be an exact solution for x, when x! = n?

Answer 1

Do you know about Gamma function that extends notion of factorial to all real and complex numbers? In general case the answer can be given only via it.

Answer 2

The Gamma function generalizes factorials. For this particular numerical question you can ask Wolfram Alpha to $$ \text{ solve } x! = 10 $$

It tells you $$ x ≈ 3.39008 $$ which makes sense: it's between $3$ and $4$.

https://www.wolframalpha.com/input/?i=solve+x!+%3D+10