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

[RSDonovan]

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?

[fiction-factory]

Hi Jim- I was thinking about the final part of the question asking if there were further solutions. I was pretty certain there can't be and thought a graph might answer that easily. I'm still not sure x=0 is a solution although I lost track when the function changed at 21:27 on thursday

[jim360]

Fairly sure that the problem is best expressed as:

find x such that Γ(x+1) = e^x

which does indeed have the second solution x=0, as Γ(1)=0!=1.

No idea what Togo's on about though.

[fiction-factory]

Thanks jim- so we lost the bit about ln(x) (for which log 0 is undefined) as RSD changed it to e^x.