Math Is Fun Forum

shocamefromebay

Member

Registered: 2007-05-30

Posts: 103

Log

Why is the base of a log always positive?
We always see

Why do we never see

?

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Log

simply because logarithms on the reals are not defined for negative numbers:

since ln(-n) is not defined for reals, neither is log_(-n)x although as you have shown there are special cases where you can define it

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if you venture into the complex realm, then you can define logarithms for negative bases:

natural logarithm of a negative real is given by the multivalued function

with principal branch:

so you could then have for some negative n

which gives for n = 5 and x = 25

and then, although this certainly is NOT '2' the expected answer, it is still of use since

Infact, you 'can' get 2 as the result if you do not use the principal branch, and instead define the logarithm as:

which gives

you can see this since:

although note that only the principal branch version is guaranteed to retain the property:

for all n,x

Last edited by luca-deltodesco (2009-02-10 08:55:31)

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The End Of All Things To Come.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Log

if you venture into the complex realm, then you can define logarithms for negative bases:

How is this?

Last edited by Identity (2009-02-10 16:38:53)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Log

if you venture into the complex realm, then you can define logarithms for negative bases:

How is this?

ln(x) is defined as being the inverse of e^x.  Because e^x can only take on positive values, ln(x) can only be defined for positive numbers.

Now going to the complex case, we now have e^z, which Euler tells us is really:

It isn't too difficult to check that e^z hits every number in the complex plain except 0.  Thus, we can define the inverse (i.e. log) there.  However, e^z isn't 1-1, specifically e^(z+2*pi*i) = e^z.  Because of this, we can only define an inverse of e^z on certain regions of the complex plane at a single time.  Things unfortunately get a bit more complicated from there, but that's the gist.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."