February 29th, 2016, 12:04 PM  #1 
Newbie Joined: Feb 2016 From: Brazil Posts: 6 Thanks: 0  Solve x! = 2^x + 8
Hi there, I have no idea how to solve this equation: $\displaystyle x! = 2^x + 8$ Cheers! Luis P. 
February 29th, 2016, 12:14 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,885 Thanks: 1504 
x = 4

February 29th, 2016, 12:29 PM  #3 
Newbie Joined: Feb 2016 From: Brazil Posts: 6 Thanks: 0  
February 29th, 2016, 12:33 PM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,885 Thanks: 1504 
... solved by observation, x = 4 works.

February 29th, 2016, 12:47 PM  #5 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 140  
February 29th, 2016, 12:54 PM  #6 
Newbie Joined: Feb 2016 From: Brazil Posts: 6 Thanks: 0  I'm sorry, maybe I wasn't clear in my question. in fact $\displaystyle 4! = 24$, then $\displaystyle 2^4=16$ and $\displaystyle 16+8=24$ Ok, the resolution is intuitive. But how should I proced in order to prove my result, isolating x, solving the equation etc? For instance: (1) $\displaystyle x*(x1)! = 2^x + 8$ (2) $\displaystyle x = \frac{2^x + 8}{(x1)!}$ (3) $\displaystyle x = \frac{2^x + 2^3}{(x1)!}$ (4) $\displaystyle x = \frac{2 * 2^{x1} + 2^3}{(x1)!}$ From that point on I dunno how to proced. 
February 29th, 2016, 01:12 PM  #7 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 140  Some equations cannot be solved in closed form. This is an example of one.

February 29th, 2016, 01:23 PM  #8 
Newbie Joined: Feb 2016 From: Brazil Posts: 6 Thanks: 0  
February 29th, 2016, 01:32 PM  #9 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
1 does not satisfy the equation because . 2 does not satisfy the equation because $\displaystyle 2!= 2\ne 2^2+ 8= 12$. 3 does not satisfy the equation because $\displaystyle 3!= 6\ne 2^3+ 8= 16$. As skeeter said, 4 does satisfy the equation: $\displaystyle 4!= 24= 2^4+ 8$. Then Mrtwhs said "Since 120=5!>25+8=40120=5!>25+8=40 and factorials grow faster than exponentials, the statement clearly has no solutions for x≥5." That answers your question. 
February 29th, 2016, 01:49 PM  #10 
Math Team Joined: Jul 2011 From: Texas Posts: 2,885 Thanks: 1504  

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