February 8th, 2017, 02:21 AM | #1 |
Member Joined: Sep 2016 From: India Posts: 88 Thanks: 30 | What is the value of $x$?
What is the value of $x$ if $x^x = x$?
Last edited by deesuwalka; February 8th, 2017 at 02:26 AM. |
February 8th, 2017, 03:31 AM | #2 |
Newbie Joined: Feb 2017 From: singapore Posts: 1 Thanks: 0 |
Suppose $\;\;$$x^{x} = x\;,\;$ where $x$ is real variable, has real solution. Equating the absolute values and taking the logarithms we get the equation $ x*\log (|x|) = \log (|x|)$ $ \therefore \;\: (x - 1)*\log (|x|) = 0$ Hence either $\;\;$$x = 1\;\;$ or $ \;\;$$\log (|x|) =0$ Therefore $\;\;$$x = 1\;\;$ or $\;\;$$|x| = 1$. Hence the solution is $\;\;$$x = 1\;\;$ or $\;\;$$x = -1$. |
February 8th, 2017, 04:47 AM | #3 | |
Senior Member Joined: Sep 2015 From: USA Posts: 1,977 Thanks: 1026 | Quote:
I don't think it is. So while $x=-1$ is in fact a solution I don't think your derivation using the absolute values is entirely valid. Last edited by romsek; February 8th, 2017 at 04:57 AM. | |