Definition of Logarithm

Date: 04/16/2001 at 08:20:23
From: Hongki, Woo
Subject: The definition of logarithm, about base

In the definition of logarithm, there are some conditions for the 
base. a^x = b iff x = log_a(b) where a > 0, a <> 1 (a is not equal to 
1), and b > 0. Why stipulate these conditions? If they are not 
assumed, are there any problems or contradictions that arise? I want 
to know why mathematicians defined logarithm this way.

Date: 04/16/2001 at 12:50:57
From: Doctor Peterson
Subject: Re: The definition of logarithm, about base

Thanks for writing to Dr. Math.

Let's just think about what happens if we violate any of these 
conditions.

First, if b <= 0, then we are trying to solve:

     a^x <= 0

You will find that there is no solution; x must be "negatively 
infinite" even to produce 0, and there is no real number x for which 
a^x < 0 if a > 0.

Next, if a = 1, we are trying to solve:

     1^x = b

Since 1^x = 1 for all x, this has no solution unless b = 1. So it 
makes no sense to talk about a logarithm with base 1.

Now, if a = 0, we have:

     0^x = b

and again, there is no solution unless b = 0. This is likewise 
useless.

Finally, if a < 0, things get tricky. We can define integral powers of 
a (which will be positive for even x and negative for odd x); but what 
about fractional powers? a^(1/2) is undefined (or rather, imaginary) 
when a is negative. If the powers are not defined for a negative base, 
then logarithms are not defined either!

As you can see, the restrictions you asked about merely keep us from 
talking about logarithms when they don't make any sense.

BUT ...

In the first and last cases, you'll notice I mentioned real and 
imaginary numbers. The fact is, your definition is valid only when we 
are considering only real numbers. We can extend the logarithm to 
apply to complex numbers, and then some of these restrictions can be 
relaxed. You can read about complex logs and some of the complications 
they introduce in the Dr. Math archives:

   The Log of a Negative Number 
   http://mathforum.org/dr.math/problems/witty3.27.98.html   

   Log of Complex Number
   http://mathforum.org/dr.math/problems/langlands.9.15.96.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/