How can I turn this into formula?

[Chitose]

First to tell, this is not my homework.

Okay, I am a writer who now try make sci-fi story, and now I need help.

I come up with method of indicator.

Basic level is 1
and level 2 are two times bigger than level 1
and level 3 are three times bigger than level 2
and level 4 are four times bigger than level 3
..and there go without limit...

.......

almost like magnitude of Earthquake right?

Can we some how turn this into formula for caculate?

like... If three times bigger than 2 is 3
than what is level that two times biggers than 2 ?

2.xxxx ?


.........

I know it's silly, but I just want to know that is there a way to create formula from this?

If it's can't, than it's okay

 

[jgens]

Let $x_1$ be the basic "level 1" and let $x_n$ denote the "n^th level." We then have that:

$$x_n = n!x_1$$

Where $n! = n(n-1)(n-2) \dots (2)(1)$

 

[lurflurf]

As jgens stated the function you want is the factorial. To generate between values (like the gamma function) you need another condition the standard one being that the function should be log convex.
antifactorial(4)~2.66403279720615568637843

 

[Chitose]

Awesome! your guy's really something.

okay, let's try replace value in formula that you gave me. (to be hornest, I'm not quit understand yet with pure formula so I wonna try.)

If I want to know exactly amont of $$x_n$$ that three times bigger than level 4.

ummm... what it going to look like?

$$x_n$$ = 3!(4) ??

confuse... :(

 

[slider142]

Awesome! your guy's really something.

okay, let's try replace value in formula that you gave me. (to be hornest, I'm not quit understand yet with pure formula so I wonna try.)

If I want to know exactly amont of $$x_n$$ that three times bigger than level 4.

ummm... what it going to look like?

$$x_n$$ = 3!(4) ??

confuse... :(

According to your description, level 4 refers to a power of 4! = 4*3*2*1 = 24 units, in whatever unit you are using to measure the phenomena. I'm just using power as an example.
So 3 times larger than level 4 has a power of 3*4! = 3*4*3*2*1 = 72 units.
If you are looking for the level that corresponds to 72, you are trying to solve the equation x! = 72. There is no well-defined way to do this, as the factorial is only defined for integer input, and its generalization, the gamma function, is not invertable. At best, you can note that 72 is between 4! and 5! and use a linear scale to approximate x as about $4 + \frac{13}{16}$.
If you meant you want the power measurement of the level that is 3 times the level of level 4, you have (4 + 3)! = 7! units.

 

[CRGreathouse]

its generalization, the gamma function, is not invertable.

It's increasing on [1.462, infinity), so it's invertable where we care about.

invgamma(x)={
  my(lx);
  if(x<1,error("x too small"));
  lx=log(x);
  solve(t=1,1e6,lngamma(t)-lx)
}

gp >invgamma(72)
%1 = 5.695574197346865243871035726

 

[Chitose]

Ah crap! I ask something impossible than!?
Sorry,

.....

So put it simple, we can calculate amount of unit but it's too complicate to reverse back to find extct 'x' level. Like we know that 3 time of level 4 is 72 units but we can't find about where that '72 units' stand between level 4 and level 5, right?

Since every 'x' level are always 'x' times bigger than previous level, formula scale are always change.

 

[lurflurf]

There is no well-defined way to do this, as the factorial is only defined for integer input, and its generalization, the gamma function, is not invertable.

Nonsense. Do you know what invertable means?

 

[lurflurf]

Ah crap! I ask something impossible than!?
Sorry,

.....

So put it simple, we can calculate amount of unit but it's too complicate to reverse back to find extct 'x' level. Like we know that 3 time of level 4 is 72 units but we can't find about where that '72 units' stand between level 4 and level 5, right?

Since every 'x' level are always 'x' times bigger than previous level, formula scale are always change.

It is not impossible at all. I would recommend a calculator or computer though as it is a lengthly calculation by hand.
We have
x!=4!*3=24*3=72
x=antifactorial(72)~4.6955741973468652438710357255

If you do not have a calculator or computer program handy you can go to http://www.wolframalpha.com/ and enter
x! == 72

 

[Chitose]

wow, ploblem slove. now I can explain how to compute this indicator in my story.

Thank you, your people are big help.