Solving equations with factorials? I looked on the internet but couldn't find anything relevant, so I was hoping you could help because I have no clue where to even start with how to solve this equations:
x! = 6

Obviously trial and error here could work, but is there any way to do it for examples where trial and error would take too long?
Thanks.
Using the Stirling approximation

$$n!\approx \sqrt{2\pi n}\left(\frac ne\right)^n$$

or in logarithms,

$$\ln(n!)\approx\frac{\ln(2\pi n)}2+n(\ln(n)-1)=\ln(N),$$

which you can solve for $n$ by numerical methods.

A crude starting approximation is

$$\frac{\ln(N)}{\ln(\ln(N))}.$$



For instance, solving for $N=14!$ yields $n=14.0022249374875\cdots$. No so bad.
The sequence of factorials is strictly increasing, so there is at most one solution.

$3!=6$, so $x=3$ is the only solution.