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 Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 06/01/2008, 03:01 PM (This post was last modified: 06/01/2008, 03:08 PM by bo198214.) Lots of ideas however mostly not working, for example: Ivars Wrote:3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b) Assume there would be a function $f$ with $f(x^y)=f(x)f(y)$ Then surely $f(x)f(x^n)=f(x^{x^n})=f((...((x^x)^x)\dots)^x)=f(x)^{n+1}$ So if there is an $x$ with $f(x)\neq 0$ we have $f(x^n)=f(x)^n$. for $y=x^n$ we also have: $f(y^{1/n})=f(x)=f(y)^{1/n}$ and both together we get $f(x^{m/n})=f(x)^{m/n}$ If we assume that $f$ is continuous then this is valid not only for rationals but also for reals $\alpha$: $f(x^\alpha)=f(x)^\alpha$. No let $x=c$ constant and let $\alpha$ be variable: $f(c^t)=f( c)^t$ $f(x)=f(c^{\log_c(x)})=f( c)^{\log_c(x)}=x^{\log_c(f( c))}=x^d$ So $f(x)=x^d$ for some constant $d$. But then: $(x^y)^d=f(x^y)=f(x)f(y)=x^d y^d$ if $d\neq 0$: $x^y=xy$ which can not be satisfied for all $x$, $y$. Hence either $d=0$ or $f(x)=0$ which we excluded in our previous considerations. Proposition. Let $f$ be a continuous function defined on the positive reals such that $f(x^y)=f(x)f(y)$ for all $x,y$, then either $f(x)=0$ for all $x$ or $f(x)=1$ for all $x$. « Next Oldest | Next Newest »

 Messages In This Thread Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/01/2008, 01:54 PM only identity maps powers to products - by bo198214 - 06/01/2008, 03:01 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/01/2008, 05:02 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by andydude - 06/02/2008, 04:46 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by bo198214 - 06/03/2008, 08:09 AM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/03/2008, 08:49 AM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by bo198214 - 06/03/2008, 01:13 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/03/2008, 01:22 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by bo198214 - 06/03/2008, 03:21 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/02/2008, 07:15 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/03/2008, 07:14 AM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/03/2008, 03:34 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by bo198214 - 06/03/2008, 05:14 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/03/2008, 09:45 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by andydude - 06/04/2008, 01:51 AM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/04/2008, 07:20 AM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by bo198214 - 06/04/2008, 02:14 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/04/2008, 02:33 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by BenStandeven - 05/12/2009, 01:00 AM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 12/04/2008, 01:15 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by JmsNxn - 04/14/2011, 11:16 PM RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by bo198214 - 04/15/2011, 07:29 AM

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