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 Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 06/02/2008, 04:46 PM (This post was last modified: 06/02/2008, 04:46 PM by andydude.) Ivars Wrote:3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b) This is where it starts to break down, because this is not a normal logarithm. Normal logarithms satisfy $\log(a^b) = b\log(a)$ Ivars Wrote:3_exp(a*b) =3_exp(a[2]b) = (3_exp(a))^(3_exp(b)) = 3_exp(a)[3]3_exp(b) Since this is not the definition of logarithms, we should really separate out the 'log' notation from the equations: $\log(a^b) = \log(a) b$ $f(a^b) = f(a) f(b)$ Using these equations we can evaluate at the following points: If $a=b^{1/b}$: $f(b) = f(b^{1/b}) f(b)$ $1 = f(b^{1/b})$ then we find that f(a) is a constant. If a=1: $f(1) = f(1) f(b)$ then f(b) is a constant. If b=1: $f(a) = f(a) f(1)$ $1 = f(1)$ then we find the point f(1)=1. If b=0: $f(1) = f(a) f(0)$ $1 = f(a) f(0)$ which means f(a) is a constant. So, not only can we show that f(x) is a constant, but we can show it in multiple ways, which is a very bad sign. I think this is proof that no such function can exist (or the only solution to all of these equations is f(x)=1). Andrew Robbins « 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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