# For What Value Of X Is X! = Ln(X)?

RSDonovan | 21:02 Thu 08th Dec 2016 | Science
For what value of x is x! = ln(x)?

from experimentation:

f 5.2903160
_1.39938e_5 less than 0

f 5.2903161
1.03394e_6 more than 0

Where f is function x! - ln(x)

So apart from x=0, there is also a result for x between 5.2903160
and 5.2903161 where f(x') = 0.

Is there a formula to calculate the value of x' exactly?

Does the value x' have any "special significance"?

I assume that after x', there are no further occurrences of x where f(x)=0?

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Blimey!

I hope someone can answer this for you. Good luck.
You can do a Taylor expansion for ln(x) but I can't immediately remember what the situation is regarding x! except that there is Stirling's approximation for large values of x

Have you tried plotting the two functions and seeing where they cross. I think they cross only once or twice. I know you say you found 2 but I am not sure what the first one was since ln(0) is undefined

x, I think but then it may be -x, but who cares?
Question Author

Where f is function x! - exp(x) i.e. e^x, where e = Euler's number.

_1.39938e_5 less than 0
and
1.03394e_6 more than 0

is it better to repost this then?
Off to bed now but will try to look in before work tomorrrow
It makes rather a big difference whether it's the logarithm or the exponential...

Also, is the factorial function here defined only for natural numbers, or has it been extended to include non-integer numbers?

The only integer answer is 0.
7 o'clock on Thursday next week.
FBG40
I always thought that logarithms were by definition exponentials, but maybe I was asleep in class at the time.
I assumed that the factorial function had covered non integer values too
Well there is a way to extend it, for sure, but it's usually written differently as the Gamma function, with Γ(n + 1) = n! and Γ(z) defined for all real numbers other than -1, -2, -3 etc.

To JD: Logarithms and exponentials are the exact opposites of each other. So log_a (a^x) = x, for all x, where a is the base of the logarithm.
hmmmmm.....
Oh, and Γ(z) is undefined for 0 as well.

I think the second solution is somewhere between 5.29 and 5.291, and I don't know if there's an exact closed form for this number or not. But I have to get on with the day.
Mathematician

A person to whom it is intuitively obvious that:-

(-½)! = sqrt(pi)
I've messed around for ages and I don't think I can either find a derive a closed solution to x' as you've defined it. It's "close" to several different combinations of some integer divided by pi to some power, but I think in each case that's luck, and anyway it's never exact.

So I'm guessing the answer to your question is "no". But it is the only positive value of x that works.
I'm going for a lie down :-)
Have you tried plotting the two functions, RSD? I'm not sure Excel can handle the Gamma function but if you have access to a maths plotter package it should be possible to do this
ff - if I understand correctly, the numerics are all sorted, ie the numerical solution for x has been found (about x=5.290316 to 7sf). The question was if there is a closed-form expression for this x, eg in terms of the square root of some integer, or pi to some power, or log of something, or the like. I don't think there is, or at least I'm not aware of it.
x2lnx2−x24+C\frac{{x}^{2}\ln{x}}{2}-\frac{{x}^{2}}{4}+C​2​​x​2​​lnx​​−​4​​x​2​​​​+C
OK, I'll bite... what?

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