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Replies: 29   Last Post: Sep 9, 2007 4:54 AM

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 Oscar Lanzi III Posts: 1,223 Registered: 12/12/04
Posted: Sep 6, 2007 6:32 PM

There is, actually, a number system in which the logarithm of -8 to the
base -2 comes out to be 3, and uniquely so.

That would be 3-adics. This is how it works: render the logarithm as
ln(-8)/ln(-2). Translate the arguments into 3-adics; each has the form
1+y where y ends with 0. Feed these into the Maclaurin series for
natural logarithms and observe that both series converge. Cancel out a
common terminal zero from the natural logarithms and do the division.
Voila, your quotient is ...000000010 which, in base ten, is 3.

In general, logarithms in p-adics may be defined for numbers ending with
1 in the p-adic representation (except p = 2 where you need ...01), and
they have all the properties of logarithms in more conventional systems.
I think it's pretty neat.

--OL